Os Grafos dos Divisores de Zero de um Anel - UBI...que muito da antiga teoria de Euclides sobre...

UNIVERSIDADE DA BEIRA INTERIOR Faculdade de Ciecircncias

Os Grafos dos Divisores de Zero de um Anel

Versatildeo final apoacutes defesa

Paulino Gregoacuterio Armando Kuebo

Dissertaccedilatildeo para obtenccedilatildeo do Grau de Mestre em

Matemaacutetica para Professores (2ordm ciclo de estudos)

Orientador Prof Doutor Celino Joseacute Martins Miguel

Covilhatilde Julho de 2018

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Dedicatoacuteria

Dedico este trabalho em especial a minha famiacutelia ao meu orientador aos meus colegas de

formaccedilatildeo agrave todos quanto contribuiacuteram para que este trabalho fosse concluiacutedo

Aos meus familiares que mesmo distante souberam compreender este momento

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Agradecimentos

Agradeccedilo agrave Deus todo poderoso pela forccedila e determinaccedilatildeo dos objetivos

Ao meu Orientador Professor Doutor Celino Joseacute Martins Miguel pelo tema que foi-me

proposto pela ajuda pelo tempo cedido que mesmo diante das suas ocupaccedilotildees sempre teve

paciecircncia em orientar-me

Agrave minha famiacutelia pela paciecircncia que sempre tiveram durante este periacuteodo de formaccedilatildeo

Ao meu querido filho Patriacutecio Pemba Kuebo que mesmo distante dele soube compreender-

me

Aos meus colegas Dr Afonso Juacutenior Lic Joaquim Dongo Kosi Lic Fernando Angelino

Policarpo Lic Joatildeo Nenkamba Lic Joseacute Massanga Lic Luiacutes Mbundo Lic Gonccedilalo Carlota

Lic Salviano Costa Lic Sebastiatildeo Lukeba Lic Mamana Combo que sempre deram-me forccedilas

e pelo encorajamento dos momentos mais difiacuteceis da minha formaccedilatildeo

Aos meus familiares Isabel Antoacutenio Pemba Manuela Maria Mateus Mendes Gregoacuterio Lau

Kuebo Paulina Armando Maria Paulo Kuebo Imaculada Mendes Domingos Mendes Olga

Chipeca e tantos outros pelas palavras de apoio e consolo

Agrave todos os Professores da Universidade da Beira Interior em particular os Professores do

Departamento de Matemaacutetica pelo apoio dado desde a nossa chegada caacute em Portugal e natildeo

soacute pelo enquadramento das diversas temaacuteticas abordadas no decorrer da nossa formaccedilatildeo

Agrave Escola Superior Politeacutecnica do ZaireSoyo em nome do Coordenador MsC Jeroacutenimo Pio

Aida e de todos os Professores desta mesma instituiccedilatildeo o meu muito obrigado

Agrave Reitoria da Universidade 11 de Novembro pelo fato de ser selecionado para o Plano de

Formaccedilatildeo de Quadros de Angola

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Resumo

Neste trabalho estudamos os grafos de divisores de zero para aneacuteis Trataremos de forma

mais exaustiva os aneacuteis comutativos uma vez que para estes aneacuteis existe um conjunto de

resultados mais abrangente No entanto no uacuteltimo capiacutetulo apresentamos alguns breves

resultados para aneacuteis natildeo comutativos Neste caso em vez de um grafo simples temos um

grafo dirigido No trabalho damos especial atenccedilatildeo ao anel dos inteiros de Gauss moacutedulo 119899

fazendo um estudo bastante completo do grafo divisor de zero para este anel

Palavras Chave

Anel grafo divisor de zero diacircmetro cintura nuacutemero cromaacutetico clique inteiros de Gauss

grafo de linha

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ix

Abstract

In this work we study zero-divisors graphs of rings We will deal more exhaustively with the

commutative rings since for these rings there is a more comprehensive set of results

However in the last chapter we present some brief results for noncommutative rings In this

case instead of a simple graph we have a directed graph In the work we give special

attention to the ring of the integers of Gauss modulo n making a fairly complete study of the

zero-divisor graph for this ring

Keywords

Ring graph zero-divisor diameter girth chromatic number click Gaussian integers line

graph

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Lista de Siacutembolos

119885[119894]- Anel dos inteiros de Gauss

119880(119877)- Grupo das unidades do anel R

119881(119866) minus Conjunto de veacutertices do grafo G

119864(119866)- Conjunto de arestas do grafo G

119889119866(119907) ou 119889119890119892(119907)- Grau do veacutertice 119907

120591(119877)- Grafo dos divisores de zero do anel R

119892(119866)- Cintura do grafo G

119889119894119886119898(119866)- Diacircmetro do grafo G

119903(119866) ou 119903119886119889(119866)- Raio do grafo G

119890(119866)- excentricidade do grafo G

119885(119877)- Conjunto dos divisores de zero do anel R

119865119902- Corpo Finito com q elementos

119885119899- Anel de inteiros moacutedulo 119899

119870119898119899 ou 119870119898119899- Grafo bipartido Completo

119886119899119899(119909)- Anulador de x

lang119899rang- Ideal principal gerado por 119899

119885119899[119894]- Anel dos inteiros Gaussianos modulo n

120574(119866)- Nuacutemero de Dominaccedilatildeo do grafo G

119870119899- Grafo Completo de n veacutertices

119888(119866)- Nuacutemero de componentes do grafo G

119871(119866)- Grafo de Linha do grafo G

120594(119866)- Nuacutemero Cromaacutetico do grafo G

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120596(119866)- Clique do grafo G

119872119899(119877)- Anel das matrizes de ordem n e de elementos no anel R

N- Nuacutemeros naturais

Z - Anel dos inteiros racionais

Q- Corpo dos racionais

R- Nuacutemeros reais

119877[119883]- Anel de polinoacutemios na indeterminada 119909 e com coeficientes em 119877

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Iacutendice Introduccedilatildeo 1

CAPIacuteTULO 1 3

BREVES NOCcedilOtildeES SOBRE ANEacuteIS 3

11 Noccedilotildees Histoacutericas 3

12 Conceitos Fundamentais da Teoria de Aneacuteis 4

CAPIacuteTULO 2 15

BREVES NOCcedilOtildeES SOBRE TEORIA DE GRAFOS 15


22 Noccedilotildees Baacutesicas da Teoria de Grafos 16

CAPIacuteTULO 3 23

O GRAFO DOS DIVISORES DE ZERO DE UM ANEL COMUTATIVO 23

31 O Grafo dos Divisor de Zero de um anel comutativo 23

32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35

O GRAFO DOS DIVISORES DE ZERO DOS INTEIROS DE GAUSS MOacuteDULO 119899 35

41 Introduccedilatildeo 35

42 GRAFO DOS DIVISORES DE ZERO PARA 119833119957119951[119842] 35

421 Grafo dos divisores de zero para Z2119899[i] 35

422 Grafo dos Divisores de Zero para 119833119954119951[119842] 119850 equiv 120785119846119848119837120786 37


43 Grafo dos Divisores de Zero para 119833119951[119842] 39

431 Quando 120533(119833119951[119842]) Eacute completo ou bipartido completo 41

432 Quando o Nuacutemero Dominante para 120533(119833119951[119842]) 120783 119848119854 120784 42

433 Quando 120649(119937119951[119946]) eacute planar 43

434 Quando 120649(119937119951[119946]) eacute regular 44

435 Quando 120649(119937119951[119946]) eacute Euleriano 45

436 Quando 120649(119937119951[119946]) eacute local 119815 46

437 Quando 120649(Zn[i]) eacute Hamiltoniano helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip47

44 O Grafo de Linha do Grafo de Divisor de Zero para o Anel de Inteiros Gaussianos Modulo 119847 51

CAPIacuteTULO 5 62

GRAFO EQUILIBRADO DOS DIVISORES DE ZERO DE ANEacuteIS DE MATRIZ 62

51 Resultados auxiliares 63

52 Resultados principais 63

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BIBLIOGRAFIA 68

1

Introduccedilatildeo

O presente trabalho estaacute composto por 5 capiacutetulos no qual alguns temas foram estudados

taxativamente no nosso trabalho e outros podem ser para futuras investigaccedilotildees acerca do

tema em causa Neste trabalho falaremos mais sobre aneacuteis comutativos e seus grafos de

divisores de zero No entanto no uacuteltimo capiacutetulo falaremos sobre aneacuteis natildeo comutativos e

respectivos grafos de divisores de zero

O estudos dos aneacuteis originou-se na teoria de polinoacutemios e da teoria de inteiros algeacutebricos O

termo anel (Zahliring) foi criado por David Hilbert em 1897 em [30]

Em 1921 Emmy Noether criou a primeira fundaccedilatildeo axiomaacutetica da teoria de aneacuteis

comutativos em [30]

Aleacutem de ser uma teoria linda e profunda por direito proacuteprio a teoria dos aneacuteis comutativos eacute

importante como base para muitos ramos da matemaacutetica

Muitos dos problemas do mundo real podem descrever-se (definir-se) na linguagem dos grafos

ou seja por intermeacutedio de uma figura que consiste num conjunto de pontos e um conjunto de

linhas que ligam alguns pares de pontos Mais geralmente uma relaccedilatildeo binaacuteria 119877 definida

sobre um conjunto 119881 pode representar-se graficamente por um conjunto de pontos que

corresponde ao conjunto 119881 e por um conjunto de arcos (ou linhas natildeo orientadas no caso da

relaccedilatildeo R ser simeacutetrica) que ligam pares de pontos 119909 119910 isin 119881 tais que 119909119877119910 Este modo de

representaccedilatildeo regista e torna evidente muitas propriedades que por vezes natildeo satildeo faacuteceis de

detectar ou explicar de outro modo

A teoria dos grafos eacute um ramo da Matemaacutetica que estuda as relaccedilotildees entre objetos de um

determinado conjunto Para tal satildeo empregados estruturas chamados de Grafos 119866(119881 119864) onde

119881 eacute um conjunto natildeo vazio de objetos denominados veacutertices e 119864 eacute um subconjunto de pares

natildeo ordenados de 119881 chamados arestas

A ideacuteia de grafo de divisor de zero de um anel comutativo foi introduzida por I Beck em [10]

onde ele estava principalmente interessado em coloraccedilatildeo Esta investigaccedilatildeo das coloraccedilotildees

de um anel comutativo foi entatildeo continuada por D Anderson e M Naseer em [5] Sua

definiccedilatildeo era ligeiramente diferente da nossa Eles deixaram todos os elementos de 119877 ser

veacutertices e fizeram 119909 119890 119910 distintos adjacentes se e somente se 119909119910 = 0 Denotamos seu grafo de

divisor de zero de 119877 por 1205910(119877) Nossos resultados para 120591(119877) tecircm anaacutelogos naturais a 1205910(119877) No

entanto sentimos que a nossa definiccedilatildeo ilustra melhor a estrutura divisor de zero de 119877

2

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CAPIacuteTULO 1

BREVES NOCcedilOtildeES SOBRE ANEacuteIS

Neste capiacutetulo expomos alguns conceitos da Teoria de Aneacuteis Procuraacutemos natildeo ser exaustivos

limitaacutemo-nos aos princiacutepios baacutesicos e algumas noccedilotildees necessaacuterias para o resto do trabalho

Daremos especial ecircnfase aos aneacuteis comutativos bem como ao anel dos inteiros de Gauss uma

vez que eacute para estes aneacuteis que faremos grande parte do estudo do grafo divisor de zero

11 Noccedilotildees Histoacutericas

A aacutelgebra comutativa eacute essencialmente o estudo dos aneacuteis que ocorrem na teoria dos nuacutemeros

algeacutebricos e geometria algeacutebrica

O assunto conhecido pela primeira vez como teoria ideal comeccedilou com o trabalho de

Richard Dedekind baseado em trabalhos anteriores de Ernst Kummer e Leopold Kronecker

Mais tarde David Hilbert apresentou o termo anel para generalizar os trabalhos anteriores

Hilbert introduziu uma abordagem mais abstrata para substituir os meacutetodos mais concretos e

computacionalmente fundamentados em coisas como anaacutelise complexa e teoria invariante

claacutessica Por sua vez Hilbert influenciou fortemente Emmy Noether que reformulou muitos

resultados anteriores em termos de uma condiccedilatildeo de cadeia ascendente agora conhecida

como a condiccedilatildeo de Noetheriano Outro marco importante foi o trabalho do estudante de

Hilbert Emanuel Lasker que introduziu ideais primaacuterios e provou a primeira versatildeo do

teorema Lasker-Noether

A figura principal responsaacutevel pelo nascimento da aacutelgebra comutativa como sujeito maduro

foi Wolfgang Krull que introduziu as noccedilotildees fundamentais de localizaccedilatildeo de um anel bem

como a dos aneacuteis locais regulares Ele estabeleceu o conceito da dimensatildeo Krull de um anel

primeiro para os aneacuteis de Noetheriano antes de avanccedilar para expandir sua teoria para cobrir

aneacuteis de valoraccedilatildeo geral e aneacuteis de Krull Esses resultados prepararam o caminho para a

introduccedilatildeo da aacutelgebra comutativa em geometria algeacutebrica uma ideacuteia que revolucionaria o

uacuteltimo assunto

Grande parte do desenvolvimento moderno da aacutelgebra comutativa enfatiza os moacutedulos

Ambos os ideais de um anel R e aacutelgebras R satildeo casos especiais de moacutedulos R de modo que a

teoria dos moacutedulos engloba a teoria ideal e a teoria das extensotildees de anel Embora jaacute tenha

sido incipiente no trabalho de Kronecker a abordagem moderna da aacutelgebra comutativa

usando a teoria dos moacutedulos geralmente eacute creditada a Krull e Noether

Os Inteiros de Gauss

4

O Matemaacutetico alematildeo Carl F Gauss produziu em todos os ramos da matemaacutetica Mas sabe-se

que sentia especial prazer pela investigaccedilatildeo em Aritmeacutetica Foi ele quem lanccedilou os

fundamentos da moderna Teoria dos Nuacutemeros em sua monumental obra Disquisitiones

Arithmeticae que conteacutem grandes contribuiccedilotildees agrave Aritmeacutetica e agrave Aacutelgebra publicada em

1801 Os inteiros de Gauss ou conjunto dos Inteiros Gaussianos satildeo nuacutemeros complexos da

forma 119886 + 119887119894 onde 119886 119890 119887 satildeo inteiros e 119894 = radicminus1 O conjunto 119885[119894] dos inteiros de Gauss surgiu

entre os anos de 1808 e 1825 eacutepoca em que o matemaacutetico Carl F Gauss investigava a

reciprocidade cuacutebica (1199093 equiv 119902 (119898119900119889 119901) onde 119901 119890 119902 119904atilde119900 119901119903119894119898119900119904) e tambeacutem a reciprocidade

biquadraacutetica (1199094 equiv 119902 (119898119900119889 119901) onde 119901 119890 119902 satildeo primos) Gauss percebeu que essa investigaccedilatildeo

se tornava mais faacutecil trabalhando em 119885[119894] o anel dos Inteiros de Gauss

Desse modo Gauss estendeu a ideia de Nuacutemero Inteiro quando definiu 119885[119894] pois descobriu

que muito da antiga teoria de Euclides sobre factoraccedilatildeo de inteiros poderia ser transportada

para esse conjunto com consequecircncias importantes para a Teoria dos Nuacutemeros

Gauss desenvolveu uma Teoria de Factorizaccedilatildeo em primos para esses nuacutemeros Complexos e

demonstrou que essa decomposiccedilatildeo em primos eacute uacutenica tal qual no Conjunto dos Nuacutemeros

Inteiros O uso desse estudo foi de fundamental importacircncia para a demonstraccedilatildeo do Uacuteltimo

Teorema de Fermat

O desenvolvimento da Teoria dos Nuacutemeros Algeacutebricos foi em parte em funccedilatildeo das tentativas

de soluccedilatildeo da equaccedilatildeo diofantina tambeacutem conhecida como equaccedilatildeo de Fermat

119909119899 + 119910 119899 = 119911119899

pois os inteiros algeacutebricos aparecem de maneira natural como ferramenta para tratar desse

assunto

Essa generalizaccedilatildeo do Conjunto dos Nuacutemeros Inteiros daacute exemplos especiais de

desenvolvimento muito mais profundos que chamamos de Teoria dos Nuacutemeros Algeacutebricos Essa

teoria eacute profunda e poderosa Aleacutem do interesse e fasciacutenio que exerce por suas proacuteprias

propriedades fornece muitas aplicaccedilotildees agrave Teoria dos Nuacutemeros que permitem uma

compreensatildeo de vaacuterios fenocircmenos antes obscuros e misteriosos

12 Conceitos Fundamentais da Teoria de Aneacuteis

Definiccedilatildeo 121 Um anel eacute um conjunto Ane empty cujos elementos podem ser adicionados e

multiplicados Um anel designa-se por (119912 + ∙) isto eacute satildeo dadas duas operaccedilotildees (119909 119910) rarr 119909 + 119910

e (119909 119910) rarr 119909 ∙ 119910 aos pares de elementos de 119860 119890119898 119860 satisfazendo as seguintes condiccedilotildees

1 Para todo 119909 119890 119910 isin 119860 temos a comutatividade da soma a saber

5

119909 + 119910 = 119910 + 119909

2 Para todo 119909 119890 119910 isin 119860 temos a associatividade da soma a saber

(119909 + 119910) + 119911 = 119909 + (119910 + 119911)

3 Existe um elemento 119890 em A tal que 119909 + 119890 = 119909 para todo 119909 isin 119860

Note 119890 = 0 Este eacute chamado elemento neutro da adiccedilatildeo

4 Para todo elemento 119909 isin 119860 existe um elemento 119910 119890119898 119860 tal que 119909 + 119910 = 0

Note 119910 = minus119909 Este eacute tambeacutem chamado de simeacutetrico de 119909

5 Para todo 119909 119910 119911 isin 119860 temos a associatividade da multiplicaccedilatildeo a saber

(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)

6 Para todo 119909 119910 119911 isin 119860 temos a distributividade da multiplicaccedilatildeo agrave direita e

esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909

Observaccedilotildees

1) Observe que a multiplicaccedilatildeo natildeo necessita ser comutativa Quando isto ocorrer

dizemos que 119860 eacute um anel comutativo

2) Um anel natildeo necessita ter elemento neutro da multiplicaccedilatildeo (isto eacute um elemento

119910 tal que 119909 ∙ 119910 = 119910 ∙ 119909 = 119909 para todo 119909 isin 119860) Este elemento se existir eacute chamado de

identidade do anel e denotado por 1 Quando um anel A possui o elemento neutro da

multiplicaccedilatildeo dizemos que 119860 eacute um anel com identidade

3) Os elementos natildeo nulos de um anel com identidade natildeo necessitam ter inversos

multiplicativos (isto eacute 119910 eacute inverso multiplicativo de x se e somente se 119909 ∙ 119910 = 119910 ∙ 119909 = 1) Os

elementos de um anel A que possuem inverso multiplicativo satildeo chamados de invertiacuteveis de A

ou unidades de A

Usaremos a notaccedilatildeo 119880(119860) = 119909 isin 119860| 119909 eacute 119906119898119886 119906119899119894119889119886119889119890 119889119890 119860

Definiccedilatildeo 122 Um elemento 119886 ne 0 de um anel A diz-se divisor de zero agrave esquerda e um

divisor de zero agrave direita se existe um elemento 119887 isin 119860 com 119887 ne 0 tal que 119886119887 = 0 119890 119887119886 = 0

respectivamente

6

Definiccedilatildeo 123 Seja (A+ ∙) um anel e seja B um subconjunto natildeo vazio de A Entatildeo B eacute um

subanel de A se e soacute se forall119909 119910 isin 119861 119904atilde119900 119904119886119905119894119904119891119890119894119905119886 119886119904 119888119900119899119889119894ccedilotilde119890119904

I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861

Exemplos O conjunto 119861 = 0 3 6 eacute um subanel de (11988512 +∙)

119861 = 0 3 6

O conjunto 119861 = 119885radic3 = 119886 + 119887radic3 119886 119887 isin 119885 eacute um subanel do anel (119877 +∙) 119877 satildeo os nuacutemeros

reais

Note que

119909 minus 119910 119909 ∙ 119910 isin 119885radic3 forall119909 119910 isin 119885radic3

119909 isin 119885radic3 119909 = 119886 + 119887radic3 119886 119887 isin 119885


119909 ∙ 119910 = (119886 + 119887radic3) ∙ (119888 + 119889radic3) = (119886119888 + 3119887119889) + (119886119889 + 119887119888)radic3

Portanto 119909 ∙ 119910 isin 119885radic3

119909 minus 119910 = (119886 + 119887radic3) minus (119888 + 119889radic3) = (119886 minus 119888) + (119887 minus 119889)radic3

Portanto 119909 minus 119910 isin 119885radic3 Logo 119885radic3 eacute 119906119898 119904119906119887119886119899119890119897 119889119890 (119877 +∙)

Definiccedilatildeo 124 Seja (A+ ∙) um anel Um subanel 119868 sub 119860 eacute um ideal de A se para cada 119886 isin

119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868

Observaccedilatildeo Um anel possui pelo menos dois ideais ele proacuteprio e o ideal formado pelo zero

do anel

Exemplo O subanel 02 eacute um ideal do anel lang1198854 +∙rang

De facto

119868 = 02 sub lang1198854 +∙rang forall119909 isin 119868 forall119886 isin 1198854 119909 ∙ 119886 isin 119868 1198854 = 0123

7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868

2 ∙ 3 = 2 isin 119868 119871119900119892119900 02 eacute 119906119898 119894119889119890119886119897 119889119890 lang1198854 +∙rang

O conjunto Q eacute um subanel de R mas natildeo eacute um ideal de R

Eacute evidente que Q subanel de R Agora note que

119909 = 2 isin 119876 119890 119886 = radic2 isin 119877 119897119900119892119900 119909 ∙ 119886 = 2radic2 notin 119876 119897119900119892119900 (119876 +∙) 119899atilde119900 eacute 119894119889119890119886119897 119889119890 119877

O conjunto lang2rang = 2119885 = 2119898 119898 isin 119885 eacute 119906119898 119894119889119890119886119897 119889119900 119886119899119890119897 (119885 +∙)

forall119909 isin 2119885 forall119886 isin 119885 119909 ∙ 119886 isin 119885

119909 isin 2119885 119909 = 2119898 119898 isin 119885

119909 ∙ 119886 = 2119898 ∙ 119886 = 2(119898 ∙ 119886) rarr 119909 ∙ 119886 = 2119898119886 119901119900119903119905119886119899119905119900 lang2rang eacute 119906119898 119894119889119890119886119897 119889119890 2119885

O conjunto 1198722 = (119886 119887119888 119889

) 119886 119887 119888 119889 isin 119877 119860 = lang1198722 +timesrang eacute um anel natildeo comutativo e com

unidade

O conjunto 119861 = (119886 1198870 0

) 119886 119887 isin 119877 eacute um subanel de A R satildeo os nuacutemeros reais Mas B natildeo eacute um

ideal de A

Uma classe importante de aneacuteis eacute apresentada na seguinte definiccedilatildeo

Definiccedilatildeo 125 Um inteiro de gauss eacute um nuacutemero complexo da forma 119886 + 119887119894 com 119886 119890 119887

inteiros cujo conjunto denotamos por 119885[119894]

Propriedades

O anel dos inteiros de Gauss tem as seguintes propriedades

8

Os elementos invertiacuteveis satildeo 1 119894 minus1 119890 minus 119894

Dois inteiros gaussianos z e w dizem-se associados se e soacute se z=wu para alguma

unidade u

Um inteiro gaussiano diz-se primo se for natildeo unidade e for divisiacutevel apenas pelos seus

associados e pelas unidades

Eacute um Domiacutenio Fatorial ou seja todo elemento tem factorizaccedilatildeo uacutenica num produto

de gaussianos primos (a menos de elementos invertiacuteveis) Note-se que alguns nuacutemeros

primos no anel dos inteiros satildeo compostos nos inteiros de Gauss por exemplo 5 =

(2 + 119894) (2 minus 119894) Os inteiros de Gauss que natildeo podem ser expressos por produto de

outros dois inteiros Gaussianos de moacutedulo maior que 1 satildeo chamados de primos de

Gauss

Pode se tornar um domiacutenio euclidiano com a norma 119873(119886 + 119887 119894) = 119886sup2 + 119887sup2

Aneacuteis Quocientes

Definiccedilatildeo 126 Seja 119877 um anel e 119868 um ideal de 119877 Defina em 119877 uma relaccedilatildeo de equivalecircncia

de forma que 119909~119910 se e soacute se 119909 minus 119910 eacute um elemento de 119868 Para o elemento 119909 a sua classe de

equivalecircncia eacute

[119909] = 119909 + 119868 Se no conjunto das classes de equivalecircncia for definida uma adiccedilatildeo por (119909 + 119868) +

(119910 + 119868) = (119909 + 119910) + 119868 e uma multiplicaccedilatildeo por (119909 + 119868)(119910 + 119868) = 119909119910 + 119868 obtemos um anel que

eacute chamado anel quociente de 119877 por 119868 e denotado por 119877119868

Definiccedilatildeo 127 Todo anel comutativo com identidade 1 ne 0 e sem divisores de zero eacute

chamado domiacutenio de integridade

Teorema 128 O conjunto dos Inteiros de Gauss eacute um Domiacutenio de integridade Em [29]

Demonstraccedilatildeo Com efeito 0 = 0 + 0119894 isin 119885[119894] Como 1 = 1 + 0119894 entatildeo 1 isin 119885[119894]

Sejam 119911 = 119886 + 119887119894 e 119908 = 119888 + 119889119894 dois Inteiros de Gauss isto eacute 119886 119887 119888 119889 isin 119885 entatildeo 119911 minus 119908 e 119911 ∙

119908 tambeacutem satildeo Inteiros de Gauss pois

119911 minus 119908 = (119886 minus 119888) + (119887 minus 119889)119894

119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908

Onde (119886 minus 119888) (119887 minus 119889) (119886119888 minus 119887119889) 119890 (119886119889 + 119888119887) satildeo inteiros Logo 119885[119894] eacute um anel comutativo com

identidade Aleacutem disso

119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎

Os inteiros Gaussianos 119885[119894] satildeo a generalizaccedilatildeo mais simples do comum inteiros 119885 e eles se

comportam da mesma maneira Em particular 119885[119894] goza de factorizaccedilatildeo uacutenica ou seja todo o

inteiro gaussiano de pode factorizar num produto de gaussianos primos de forma uacutenica (a

menos de multiplicaccedilatildeo por unidades) e isso nos permite trabalhar sobre 119885[119894] da mesma

maneira que fazemos sobre 119885 Fazemos isso porque 119885[119894] eacute o lugar natural para estudar certas

propriedades de 119885 Em particular eacute o melhor lugar para examinar somas de dois quadrados

porque em 119885[119894] podemos factorizar uma soma de dois quadrados inteiros em fatores lineares

1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma

Definiremos tambeacutem uma funccedilatildeo muito importante na aritmeacutetica desse conjunto que eacute

chamada de Norma onde

Definiccedilatildeo 129 Para 119911 = 119886 + 119887119894 isin 119885[119894] a Norma eacute o produto

119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872

Pensando em 119886 + 119887119894 como um nuacutemero complexo sua Norma eacute o quadrado de seu

moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2

A razatildeo pela qual preferem lidar com Normas em 119885[119894] em vez de valores absolutos eacute que as

Normas satildeo inteiros (em vez de raiacutezes quadradas) e as propriedades de divisibilidade em 119885

vatildeo fornecer informaccedilotildees importantes sobre as propriedades de divisibilidade em 119885[119894] Isto eacute

baseado na seguinte propriedade algeacutebrica da Norma Em [29]

Teorema 1210 A Norma eacute multiplicativa ou seja 119911 ∙ 119908 = 119911 ∙

Demonstraccedilatildeo Fazendo 119911 = 119886 + 119887119894 119890 119908 = 119888 + 119889119894 Entatildeo 119911 ∙ 119908 = (119886 + 119887119894) ∙ (119888 + 119889119894)

Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)

Verificamos que 1 e 2 tem o mesmo resultados logo

119873(119911119908) = 119873(119911)119873(119908) ∎

Determinaremos a seguir os inteiros de Gauss que tem inversos multiplicativos em 119885[119894]

Divisibilidade e primos em 119885[119894] e 119885

119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872

eacute mais uacutetil na teoria dos nuacutemeros do que o valor absoluto porque a norma eacute sempre um

nuacutemero inteiro comum A propriedade multiplicativa da norma implica que se um inteiro

Gaussiano 120572 divide um inteiro Gaussiano 120574 isto eacute se

120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)

Por isso as questotildees sobre divisibilidade em 119885[119894] geralmente se reduzem a questotildees sobre

divisibilidade em 119885

Teorema 1211 Factorizaccedilatildeo prima em 119885[119894] Qualquer inteiro Gaussiano natildeo nulo e natildeo

unidade eacute factorizado em um produto de Gaussianos primos A prova eacute semelhante agrave prova

em 119885 Em [29]

Demonstraccedilatildeo Considere qualquer inteiro Gaussiano 120574 Se 120574 em si eacute um primo Gaussiano

entatildeo terminamos Se natildeo entatildeo 120574 = 120572120573 para alguns 120572 120573 isin 119885[119894] com menor norma Se 120572 120573

natildeo satildeo ambos primos Gaussianos factorizamos em Gaussiano inteiros de norma ainda menor

e assim por diante Esse processo deve terminar desde as normas sendo nuacutemeros naturais

natildeo podem diminuir para sempre Por conseguinte eventualmente obter uma factorizaccedilatildeo

prima Gaussiana de 120574 ∎

Como em 119885 natildeo eacute imediatamente claro que a factorizaccedilatildeo prima eacute uacutenica No entanto vemos

nesta secccedilatildeo que a factorizaccedilatildeo prima uacutenica deteacutem em 119885[119894] pelas mesmas razotildees que em 119885

Conjugados

O conjugado de 119911 = 119886 + 119887119894 eacute 119911 = 119886 minus 119887119894 As propriedades baacutesicas da conjugaccedilatildeo

11

(natildeo apenas em 119885[119894] mas para todos os nuacutemeros complexos z) satildeo

i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112

Estes podem ser verificados escrevendo 1199111 = 1198861 + 1198871119894 1199112 = 1198862 + 1198872119894 e trabalhando ambos os

lados de cada identidade Usamos essas propriedades de conjugaccedilatildeo para que decirc o primeiro

passo para uma classificaccedilatildeo de primos Gaussianos

Teorema 1212 Primos Gaussianos Reais Um primo comum 119901 isin 119873 eacute um primo Gaussiano

hArr 119901 natildeo eacute a soma de dois quadrados (E obviamente 119901 lt 0 eacute um primo Gaussiano hArr minus119901 isin 119873

eacute um primo Gaussiano) Em [29]

Demonstraccedilatildeo (lArr) Suponhamos que tenhamos um 119901 primo comum que natildeo eacute um primo

Gaussiano de modo que se divide em 119885 [119894]

119901 = (119886 + 119887119894) 120574

onde 119886 + 119887119894 119890 120574 satildeo nuacutemeros inteiros Gaussianos com 119873(119886 + 119887119894) lt 1199012 e 119873(120574) lt 1199012 e tambeacutem

119873(119886 + 119887119894) gt 1 119890 119873(120574) gt 1 Tomando conjugados de ambos os lados obtemos

119901 = (119886 minus 119887119894)

uma vez que 119901 eacute 119903119890119886119897 e portanto 119901 = Multiplicando estas duas expressotildees para 119901 daacute

1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2

onde tanto 1198862 + 1198872 | 120574 |2 gt 1 Mas a uacutenica tal factorizaccedilatildeo de 1199012 eacute 119901119901 portanto

119901 = 1198862 + 1198872

(rArr) Inversamente se um primo 119901 comum eacute igual 1198862 + 1198872 com 119886 119887 isin 119885 entatildeo 119901 natildeo eacute um

primo Gaussiano porque tem a factorizaccedilatildeo Gaussiana primo

119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎

Observe tambeacutem que os fatores 119886 minus 119887119894 119890 119886 + 119887119894 de 119901 satildeo primos Gaussianos porque a sua

norma eacute o nuacutemero primo 1198862 + 1198872 = 119901 Aleacutem disso todos os Gaussianos primos 119886 +

12

119887119894 119900119899119889119890 119886 119887 ne 0 vecircm em pares conjugados como este Isto eacute entatildeo porque se um membro do

par se factorizar em 120572120573 entatildeo o seu conjugado eacute fatorizado em 120572120573

O que ainda natildeo estaacute claro eacute se todos os primos Gaussianos 119886 + 119887119894 119888119900119898 119886 119887 diferentes de zero

satildeo fatores de primos comuns 119901 = 1198862 + 1198872 Eacute concebiacutevel que 119886 + 119887119894 pode ser um primo

Gaussiano enquanto 1198862 + 1198872 eacute um produto de dois ou mais primos comuns Nesta Seccedilatildeo

descartamos isso com a ajuda de uma uacutenica factorizaccedilatildeo prima em 119885[119894]

De qualquer forma podemos ver que esclarecimentos adicionais sobre a natureza dos primos

Gaussianos depende de encontrar outra maneira de descrever os primos comuns que satildeo

somas de dois quadrados Os primos que natildeo satildeo somas de dois quadrados satildeo da forma 4119899 +

3 O complemento a este resultado que qualquer primo da forma 4119899 + 1 eacute uma soma de dois

quadrados e um famoso teorema descoberto por Fermat

Divisatildeo em 119885[119894]

A factorizaccedilatildeo prima uacutenica em 119885[119894] como em 119885 depende do algoritmo euclidiano que

depende por sua vez

Teorema 1213 Propriedade de divisatildeo de 119885[119894] 119878119890 120572 120573 ne 0 estiverem em 119885[119894] entatildeo

existem 120583 120588 pertencentes a Z[i] e tal que

120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |

Demonstraccedilatildeo Esta propriedade torna-se oacutebvia uma vez que se vecirc que o Gaussiano muacuteltiplo

inteiro 120583120573 de qualquer inteiro Gaussiano 120573 ne 0 forma uma grade quadrada em o plano

complexo Isso ocorre porque a multiplicaccedilatildeo de 120573 por 119894 roda o vetor de 0 119886 120573 atraveacutes de 90deg

portanto 0 120573 119890 119894120573 satildeo trecircs cantos de um quadrado Todos os outros muacuteltiplos de 120573 satildeo somas

(ou diferenccedilas) de 120573 119890 119894120573 portanto eles estatildeo no cantos de uma grade quadrada (Figura 1)

Em [29]

13

Figura 1 Muacuteltiplos dos Inteiros Gaussianos

Qualquer nuacutemero inteiro Gaussiano 120572 estaacute em um desses quadrados e haacute um canto mais

proacuteximo 120583120573 (natildeo necessariamente uacutenico mas natildeo importa) Entatildeo

120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900

entatildeo | ρ | eacute menor do que o lado de um quadrado a saber | β | ∎

Graccedilas agrave propriedade da divisatildeo que temos

1 Um algoritmo euclidiano 119901119886119903119886 119885[119894]

2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]

3 A propriedade do divisor primo se um 120587 primo divide 120572120573 entatildeo 120587 divide

120572 119900119906 120587 119889119894119907119894119889119890 120573

Como uma primeira aplicaccedilatildeo de factorizaccedilatildeo prima uacutenica em 119885[119894] completamos a descriccedilatildeo

dos primos Gaussianos Laacute encontramos que os primos reais Gaussianos satildeo primos comuns que

natildeo satildeo somas de dois quadrados e seus negativos Tambeacutem eacute claro que o Gaussiano

imaginaacuterio puro os primos satildeo da forma plusmn 119894119901 onde 119901 eacute um primo Gaussiano real Assim ele

permanece para descrever os primos Gaussianos 119886 + 119887119894 com 119886 119887 diferentes de zero

Teorema 1214 Primos Gaussianos Imaginaacuterios Os primos Gaussianos 119886 + 119887119894 119888119900119898 119886 119887

diferentes de zero satildeo fatores de primos comuns 119901 da forma 1198862 + 1198872 Em [29]

Demonstraccedilatildeo Primeiro como observado na Seccedilatildeo anterior se 119886 + 119887119894 eacute um primo Gaussiano

entatildeo eacute 119886 minus 119887119894 (119901119900119903119902119906119890 119904119890 119886 minus 119887119894 = 120572120573 natildeo eacute primo nem eacute 119886 + 119887119894 = 120572120573)

Em seguida (119886 minus 119887119894) (119886 + 119887119894) eacute uma factorizaccedilatildeo Gaussiana prima (necessariamente uacutenica)

do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)

Mas 119901 deve ser um primo comum Na verdade se

119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885

entatildeo os fatores primos Gaussianos de 119903 119890 119904 datildeo uma factorizaccedilatildeo Gaussiana prima de 119901

diferente de (119886 minus 119887119894) (119886 + 119887119894) (ou dois fatores reais 119903 119890 119904 119900119906 ge quatro fatores complexos) ∎

15

CAPIacuteTULO 2

BREVES NOCcedilOtildeES SOBRE TEORIA DE GRAFOS

Neste capiacutetulo expomos alguns conceitos da Teoria de Grafos definiccedilotildees e alguns exemplos

Apenas definimos as noccedilotildees necessaacuterias para o trabalho Para um estudo mais profundo da

teoria dos grafos o leitor pode consultar [21 36 48]


Podemos dizer como Harary que a teoria dos grafos foi redescoberta muitas vezes ou

entatildeo que problemas do interesse de diversas aacutereas foram estudados separadamente e

mostraram caracteriacutesticas semelhantes Importante de qualquer modo eacute observar que o

periacuteodo transcorrido entre a demonstraccedilatildeo de Euler sobre o problema das sete pontes de

Konigsberg e a uacuteltima deacutecada do seacuteculo XIX - mais de 150 anos viu apenas o surgimento de

alguns poucos trabalhos Assim eacute que em 1847 Kirchhoff utilizou modelos de grafos no

estudo de circuitos eleacutectricos e ao fazecirc-lo criou a teoria das aacutervores - uma classe de grafos

para caracterizar conjuntos de ciclos independentes Dez anos mais tarde Cayley seguiria a

mesma trilha embora tendo em mente outras aplicaccedilotildees dentre as quais se destaca a

enumeraccedilatildeo dos isocircmeros dos hidrocarbonetos alifaacuteticos saturados em quiacutemica orgacircnica

Enfim Jordan (1869) se ocupou tambeacutem das aacutervores de um ponto de vista estritamente

matemaacutetico

Muitos eventos que provaram ser importantes satildeo relacionados com problemas com pouca

aplicaccedilatildeo praacutetica Hamilton em 1859 inventou um jogo que consistia na busca de um

percurso fechado envolvendo todos os veacutertices de um dodecaedro regular de tal modo que

cada um deles fosse visitado uma uacutenica vez Eacute interessante aliaacutes observar que os problemas

de Hamilton e de Euler encontraram aplicaccedilatildeo respectivamente um e dois seacuteculos mais

tarde no campo da pesquisa operacional Kempe (1879) procurou sem sucesso demonstrar a

conjectura das quatro cores apresentada por Guthrie a De Morgan provavelmente em

1850 Este problema um dos mais importantes jaacute abordados pela teoria dos grafos oferece

interesse apenas teoacuterico trata-se de provar que todo mapa desenhado no plano e dividido em

um nuacutemero qualquer de regiotildees pode ser colorido com um maacuteximo de quatro cores sem que

duas regiotildees fronteiriccedilas recebam a mesma cor Taity (1880) divulgou tambeacutem uma prova

infelizmente baseada numa conjectura falsa e Heawood (1890) mostrou que a prova de

Kempe estava errada obtendo no processo uma prova vaacutelida para 5 cores a prova para 4

cores somente foi obtida em 1976 A importacircncia do problema reside nos desenvolvimentos

teoacutericos trazidos pelas tentativas de resolvecirc-lo as quais enriqueceram a teoria dos grafos em

diversos recursos ao longo da primeira metade do seacuteculo XX exemplificando Birkhoff (1912)

16

definiu os polinoacutemios cromaacuteticos Whitney (1931) criou a noccedilatildeo de grafo dual e Brooks (1941)

enunciou um teorema fornecendo um limite para o nuacutemero cromaacutetico de um grafo

Outros eventos importantes podem ser citados Menger (1926) demonstrou um importante

teorema sobre o problema da desconexatildeo de itineraacuterios em grafos e Kuratowski (1930)

encontrou uma condiccedilatildeo necessaacuteria e suficiente para a planaridade de um grafo Turaacuten

(1941) foi o pioneiro do ramo conhecido como teoria extremal de grafos e Tutte (1947)

resolveu o problema da existecircncia de uma cobertura minimal em um grafo Vale a pena

registrar que o termo grafo foi usado pela primeira vez por Sylvester em 1878 e que o

primeiro livro especiacutefico sobre grafos foi publicado por Konig em 1936 uma eacutepoca na qual

conforme Wilder o assunto era considerado um campo morto

A partir de 1956 com a publicaccedilatildeo dos trabalhos de Ford e Fulkerson (1956) Berge (1957) e

Ore (1962) o interesse pela teoria dos grafos comeccedilou a aumentar crescendo rapidamente

no mundo todo conforme cita Harary em 1969 foi publicada por J Turner A imensa maioria

dos livros sobre grafos foi publicada depois de 1970 em grande parte sob a influecircncia das

obras de Berge e Harary O desenvolvimento dos computadores levou agrave publicaccedilatildeo de vaacuterias

obras dedicadas aos algoritmos de grafos abrindo assim possibilidades crescentes de

utilizaccedilatildeo aplicada da teoria

22 Noccedilotildees Baacutesicas da Teoria de Grafos

Nesta seccedilatildeo falaremos sobre os grafos natildeo orientados

Definiccedilatildeo 221 Designa-se por grafo (natildeo orientado) um terno 119866 = (119881(119866) 119864(119866) 120595(119866)) onde

119881 = 119881(119866) eacute um conjunto natildeo vazio 119864 = 119864(119866) eacute um conjunto disjunto de 119881 119890 120595119866 eacute uma funccedilatildeo

tal que para cada 119890 isin 119864 120595119866(119890) denota um par natildeo ordenado de elementos (natildeo

necessariamente distintos) de 119881 Neste caso 119881 designa-se por conjunto de Veacutertices 119864 por

conjunto de arestas e 120595119866 por funccedilatildeo de incidecircncia

Definiccedilatildeo 222 (Grafo simples) Um grafo diz-se simples se natildeo conteacutem arestas paralelas

nem lacetes

Definiccedilatildeo 223 Dado um grafo 119866 simples designa-se por grafo complementar de 119866 e

denota-se por 119866119862 um grafo simples cujo conjunto de veacutertices eacute 119881(119866) e no qual dois veacutertices

satildeo adjacentes se e soacute se natildeo satildeo adjacentes em 119866

17

Figura 2 Grafo 119866 figura 3 Grafo 119866119862

Definiccedilatildeo 224 Dois grafos 119866 = (119881(119866) 119864(119866) 120595(119866)) e 119867 = (119881(119867) 119864(119867) 120595(119867)) dizem-se

isomorfos denotando-se esta relaccedilatildeo de isomorfismo por 119866 cong 119867 se existem duas bijeccedilotildees

120593 119881(119866) rarr 119881(119867) 119890 120579 119864(119866) rarr 119864(119867) tais que

120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)

Por outras palavras dois grafos dizem-se isomorfos se existe uma bijeccedilatildeo entre os respetivos

conjuntos de veacutertices e uma bijeccedilatildeo entre os respetivos conjunto de arestas que preservam as

relaccedilotildees de adjacecircncia e de incidecircncia

Definiccedilatildeo 225 Designa-se por isomorfismo entre dois grafos simples 119866 119890 119867 uma bijeccedilatildeo

120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)

De acordo com as Definiccedilotildees 24 e 25 podemos concluir que dois grafos satildeo isomorfos

quando existe um isomorfismo entre eles

Aresta Incidente eacute aquela que liga dois veacutertices distintos

Arestas Adjacentes satildeo aquelas que estatildeo ligadas a um mesmo veacutertice e natildeo satildeo arestas

muacuteltiplas

Veacutertices Adjacentes satildeo aqueles que estatildeo ligados por uma mesma aresta

Ao nuacutemero de veacutertices de um grafo 119866 chamamos de ordem de 119866 que indicamos por 119907(119866) ou

apenas 119907 e ao nuacutemero de arestas de um grafo 119866 chamamos dimensatildeo de 119866 que indicamos

por 휀(119866) ou apenas 휀

18

Definiccedilatildeo 226 Dado um grafo 119866 e um veacutertice 119907 isin 119881(119866) designamos por grau de 119907 o nuacutemero

de arestas incidentes no veacutertice 119907 que indicamos por 119889119866(119907) O maior grau dos veacutertices de 119866

indicamos por ∆(119866) e o menor grau dos veacutertices de 119866 indicamos por 120575(119866)

Caminho eacute uma sucessatildeo de veacutertices e arestas tal que cada aresta liga o veacutertice que a

precede ao veacutertice que a segue natildeo repetindo arestas

Figura 4

Os veacutertices A B C e D representam um caminho neste grafo

Caminho Fechado eacute aquele que comeccedila e termina no mesmo veacutertice

Figura 5

Ciclo eacute um caminho fechado

Figura 6

Passeio eacute um caminho onde pode haver repeticcedilatildeo de arestas e de veacutertices

19

Figura 7

Ponte eacute uma aresta cuja remoccedilatildeo reduz a conexidade do grafo

Exemplo ℎ 119890 119894 satildeo as pontes do grafo

Figura 8

Subgrafo de um Grafo G eacute aquele cujo o conjunto dos veacutertices e o conjunto das arestas satildeo

subconjuntos do conjunto de veacutertices e de arestas respetivamente de 119866

20

Figura 9 Grafo 119866 e Figura 10 Subgrafo de 119866

Definiccedilatildeo 227 Dado um grafo 119866 eliminando todos os lacetes e substituindo cada conjunto

de arestas paralelas por uma uacutenica aresta obtemos um subgrafo abrangente de 119866 ao qual

chamamos de subgrafo de suporte de arestas

Grafo completo dizemos que 119866 eacute um grafo completo quando todos os veacutertices satildeo

adjacentes

Grafo Bipartido eacute aquele em que o conjunto dos seus veacutertices admite uma particcedilatildeo 1198811 1198812

de tal maneira que toda a aresta de G une um veacutertice de 1198811 a um veacutertice de 1198812

Figura 11

Nota Um grafo eacute bipartido se e soacute se natildeo tem circuitos de comprimento iacutempar

Definiccedilatildeo 228 Dizemos que 119866 eacute um grafo conexo se para cada par de veacutertices existe

sempre um caminho que os une

Grafo Desconexo eacute aquele que natildeo eacute conexo

Componentes Conexas de um grafo desconexo satildeo subgrafos conexos disjuntos em relaccedilatildeo

aos veacutertices e maximais em relaccedilatildeo agrave inclusatildeo

Floresta eacute um grafo cujas componentes conexas satildeo aacutervores

Figura 12

21

Definiccedilatildeo 229 Chamamos cintura do 119866 ao comprimento do ciclo de menor comprimento de

119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin

Excentricidade Seja G um grafo e 119907 um veacutertice entatildeo a maior distacircncia entre 119907 e todos os

outros veacutertices de G designa-se por excentricidade de 119907 e denota-se por 119890119866(119907) 119900119906 119890(119907)

Diacircmetro Dado um grafo G a maior excentricidade dos seus veacutertices designa-se por

diacircmetro

Raio eacute a distacircncia miacutenima de todos os veacutertices G

Centro satildeo os veacutertices onde as excentricidades satildeo mais pequenas

Exemplo Dado o grafo 119866 indica o diacircmetro o raio a excentricidade e a cintura do grafo

completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1

Grafo Regular eacute um grafo no qual todos os veacutertices tecircm o mesmo grau

Grafo Planar eacute um grafo que pode ser imerso no plano de tal forma que suas arestas natildeo se

cruzam

Grafo Estrela eacute um grafo onde existe um veacutertice central que eacute adjacente a todos os outros

veacutertices do grafo

Grafo Euleriano um grafo G eacute Euleriano se e somente se G eacute conexo e cada veacutertice de G tem

grau par

22

Grafo Hamiltoniano um grafo G eacute dito ser Hamiltoniano se existe um ciclo em G que

contenha todos os seus veacutertices sendo que cada veacutertice soacute aparece uma vez no ciclo

23

CAPIacuteTULO 3

O GRAFO DIVISOR DE ZERO DE UM ANEL COMUTATIVO

Neste capiacutetulo trataremos da noccedilatildeo do grafo dos divisores de zero Este eacute o tema principal do

trabalho Apresentamos definiccedilotildees e demonstraccedilotildees de alguns teoremas importantes

Apresentamos tambeacutem alguns exemplos

31 O Grafo Divisor de Zero de um anel comutativo

Seja 119877 um anel comutativo com identidade e seja 119885 (119877) seu conjunto de divisores de zero

Associamos um grafo (simples) 120591(119877) 119886 119877 119888119900119898 119907eacute119903119905119894119888119890119904 119885(119877)lowast = 119885 (119877) minus 0 isto eacute 119881(120591(119877)) =

119885(119877)lowast conjunto de divisores zero natildeo nulos de 119877 e para distintos 119909 119910 isin 119885(119877)lowast Os veacutertices 119909 119890 119910

satildeo adjacentes se e somente se 119909119910 = 0 Assim 120591(119877) eacute o grafo vazio se e somente 119904119890 119877 eacute um

domiacutenio integridade

Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24

O principal objetivo deste capiacutetulo eacute estudar a interaccedilatildeo das propriedades teoacutericas de 119877 com

as propriedades teoacutericas de 120591(119877) Este estudo ajuda a iluminar a estrutura de 119877

119875119886119903119886 119909 119910 isin 119885lowast (119877) 119889119890119891119894119899119886 119909 ~ 119910 119904119890 119909119910 = 0 119900119906 119909 = 119910 A relaccedilatildeo eacute sempre reflexiva e simeacutetrica

mas natildeo usualmente transitiva O grafo de divisor de zero 120591(119877) mede esta falta de

transitividade no sentido de que ~ eacute transitiva se e somente se 120591(119877) estiver completo





veacutertices e 119909 119890 119910 satildeo adjacentes se e somente se 119909119910 = 0 Denotamos seu grafo de divisor de

zero de 119877 por 1205910(119877) Em 1205910(119877) o veacutertice 0 eacute adjacente a cada outro veacutertice Nossos resultados

para 120591(119877) tecircm anaacutelogos naturais a 1205910(119877) No entanto sentimos que a nossa definiccedilatildeo ilustra

melhor a estrutura divisor de zero de 119877

Na seccedilatildeo 32 damos muitos exemplos mostramos que 120591(119877) eacute conexo e 119889119894119886119898 (120591(119877)) le 3 e

determinamos quando 120591(119877) eacute um grafo completo ou um grafo estrela Um passo-chave eacute

caracterizar quando um veacutertice eacute adjacente a todos os outros veacutertices Na terceira seccedilatildeo

estudamos o grupo de automorfismo do 120591(119877)

Incluiremos definiccedilotildees baacutesicas da teoria dos grafos conforme necessaacuterio Referecircncias baacutesicas

para a teoria de grafos satildeo [21 28 36] Para a teoria dos aneacuteis comutativos ver [9 32 34]

Todos os aneacuteis 119877 satildeo comutativos com identidade excepto no uacuteltimo capiacutetulo Como de

costume os aneacuteis de inteiros e 119894119899119905119890119894119903119900119904 119898119900119889119906119897119900 119899 seratildeo denotados por 119885 119890 119885119899

respectivamente e 119865119902 seraacute o corpo finito com 119902 elementos

32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)

33 Propriedades de 120533(119825)

Nesta seccedilatildeo mostramos que 120591(119877) eacute sempre conexo e tem diacircmetro menor ou igual a 3

Determinamos quais grafos completos e grafo estrela podem ser realizados como 120591(119877)

Comeccedilamos com alguns exemplos que motivam resultados posteriores

Exemplo (a) Abaixo estatildeo os grafos de divisor de zero para vaacuterios aneacuteis Observe que esses

exemplos mostram que aneacuteis natildeo-isomorfos podem ter o mesmo grafo de divisor de zero e

que o grafo de divisor zero natildeo deteta elementos nilpotente

1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26

(b) Na parte a) acima todos os grafos conexos com menos de quatro veacutertices podem ser

realizados como 120591(119877) Dos onze grafos com quatro veacutertices apenas seis satildeo conexos Destes

seis apenas os trecircs grafos seguintes podem ser realizados como 120591(119877)

1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0

Tabela da multiplicaccedilatildeo

times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906

Observaccedilatildeo

1199062 + 119906 + 1 = 0 harr 1199062 = minus119906 minus 1 harr 1199062 = 119906 + 1

1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)

Em seguida esboccedilamos uma prova de que o 120591 grafo com veacutertices 119886 119887 119888 119889 e arcos

119886 minus 119887 119887 minus 119888 119888 minus 119889 natildeo pode ser realizado como 120591(119877)

Figura 20

Suponha que existe um anel 119877 com 119885(119877) = 0 119886 119887 119888 119889 e acima das relaccedilotildees de divisores de

zero Entatildeo 119886 + 119888 isin 119885 (119877) uma vez que (119886 + 119888) 119887 = 0 Portanto 119886 + 119888 deve ser

0 119886 119887 119888 119900119906 119889 Uma simples verificaccedilatildeo produz 119886 + 119888 = 119887 como a uacutenica possibilidade

Similarmente 119887 + 119889 = 119888 Daiacute 119887 = 119886 + 119888 = 119886 + 119887 + 119889 Assim 119886 + 119889 = 0 Assim 119887119889 =

119887 (minus119886) = 0 uma contradiccedilatildeo As provas dos outros dois grafos conexos natildeo-realizaacuteveis em

quatro veacutertices satildeo semelhantes

(c) Vimos acima que 120591(119877) pode ser um triacircngulo ou quadrado Mas 120591(119877) natildeo pode ser um n-gon

para qualquer 119899 ge 5 (As provas satildeo semelhantes agravequela da parte (b) acima Isto tambeacutem se

segue diretamente dos Teoremas 331 e 334) No entanto para cada 119899 ge 3 haacute um divisor

de zero de grafo com um ciclo 119899 Para 119877119899 =1198852[1198831hellip119883119899]

119868 onde 119868 = (1198831

2 hellip 1198831198992 11988311198832 hellip 1198831198991198831) Entatildeo

120591(119877119899) eacute finito e tem um ciclo de comprimento 119899 ou seja 1198831 minus 1198832 minus ⋯ minus ⋯ 119883119899 minus 1198831

Sejam 119860 119890 119861 domiacutenios de integridades e seja 119877 = 119860 times 119861 Entatildeo 120591(119877) eacute um grafo bipartido

completo (isto eacute 120591(119877) pode ser dividido em dois conjuntos de veacutertices disjuntos 1198811 =

(119886 0) ∶ 119886 isin 119860lowast e 1198812 = (0 119887) ∶ 119887 isin 119861lowast e dois veacutertices 119909 119890 119910 satildeo adjacentes se e somente se

estiverem em conjuntos de veacutertices distintos) com |120591(119877)| = |119860| + |119861| minus 2 O grafo bipartido

29

completo com conjuntos de veacutertices com 119898 119890 119899 elementos respetivamente seraacute denotado

por 119870119898119899 Um grafo bipartido completo da forma 1198701119899 eacute chamado de grafo de estrela Se 119860 =

1198852 entatildeo 120591(119877) eacute um grafo em estrela |120591(119877)| = |119861| Por exemplo 120591(119865119901 times 119865119902) = 119870119901minus1119902minus1 e

120591(1198852 times 119865119902) = 1198701119902minus1 Damos dois exemplos especiacuteficos

1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

Naturalmente 120591(119877) pode ser infinito (isto eacute um anel pode ter um nuacutemero infinito de

divisores de zero) Mas 120591(119877) eacute provavelmente de maior interesse quando eacute finito pois entatildeo

se pode desenhar 120591(119877) Vamos afirmar a maioria dos resultados em um cenaacuterio tatildeo geral

quanto possiacutevel e em seguida muitas vezes se especializam para o caso finito Em seguida

mostramos que 120591(119877) eacute finito (exceto caso trivial quando 120591(119877) estaacute vazio) somente quando 119877 eacute

ele mesmo finito Assim muitas vezes restringiremos o caso em que 119877 eacute um anel finito

Lembre-se que se 119877 eacute finito entatildeo cada elemento de 119877 eacute uma unidade ou um divisor de zero

cada ideal primo de 119877 eacute um aniquilador ideal e cada divisor de zero de 119877 eacute nilpotente se e

somente se 119877 eacute local Aleacutem disso se 119877 eacute um anel local finito com 119872 ideal maximal entatildeo

|119877 | = 119901119899 para algum primo 119901 e inteiro 119899 ge 1 Entatildeo |120591(119877)| = 119901119898 minus 1 para algum inteiro 119898 ge

0 A essecircncia de nosso primeiro resultado eacute que 119885(119877) eacute finita se e somente se 119877 eacute finito ou um

domiacutenio integridade (este resultado com uma prova diferente e o fato de que |119877| le

|119885(119877)|2 119902119906119886119899119889119900 2 le |119885(119877)|infin satildeo devidas a N Ganesan [17 119879119890119900119903119890119898119886 1] em [32] para anaacutelogos

natildeo comutativos)

Teorema 331 Seja 119877 um anel comutativo Entatildeo 120591(119877) eacute finito e somente se 119877 eacute finito ou

um domiacutenio de integridade Em particular se 1 le |120591(119877)| lt infin entatildeo 119877 eacute finito e natildeo um corpo

Em [36]

Demonstraccedilatildeo Suponha que 120591(119877)(= 119885(119877)lowast) seja finito e natildeo vazio Entatildeo existe 119909 119910 isin 119877 natildeo

nulos com 119909119910 = 0 Seja 119868 = 119886119899119899 (119909) Entatildeo 119868 sub 119885 (119877) eacute finito e 119903119910 isin 119868 para todo 119903 isin 119877 Se 119877 eacute

infinito entatildeo existe um 119894 isin 119868 com 119895 = 119903 isin 119877|119903119910 = 119894 infinito Para qualquer 119903 119904 isin

119895 (119903 minus 119904)119910 = 0 entatildeo 119886119899119899(119910) sub 119885(119877) eacute infinito uma contradiccedilatildeo Assim 119877 deve ser finito ∎

Teorema 332 Seja 119877 um anel comutativo Entatildeo 120591(119877) eacute conexo e 119889119894119886119898(120591(119877)) le 3 Aleacutem

disso se 120591(119877) conteacutem um ciclo entatildeo 119892(120591(119877)) le 7 Em [36]

Demonstraccedilatildeo Seja 119909 119910 isin 119885(119877)lowast distintos Se 119909119910 = 0 entatildeo 119889 (119909 119910) = 1 Entatildeo suponha que

119909119910 seja diferente de zero Se 1199092 = 119910 2 = 0 entatildeo 119909 minus 119909119910 minus 119910 eacute um caminho de comprimento

2 assim 119889 (119909 119910) = 2 Se 1199092 = 0 119890 1199102 ne 0 entatildeo haacute um 119887 isin 119885 (119877)lowast minus (119909 119910) com 119909119910 = 0 Se

119887119909 = 0 entatildeo 119909 minus 119887 minus 119910 eacute um caminho de comprimento 2 Se 119887119909 ne 0 entatildeo 119909 minus 119887119909 minus 119910 eacute um

caminho de comprimento 2 Em ambos os casos 119889(119909 119910) = 2 Um argumento semelhante eacute

vaacutelido se 119910 2 = 0 119890 1199092 ne 0 Assim podemos assumir que 119909119910 1199092 119890 1199102 satildeo todos diferentes de

zero Portanto haacute um 119886 119887 isin 119885(119877)lowast minus 119909 119910 119888119900119898 119886119909 = 119887119910 = 0 Se 119886 = 119887 entatildeo 119909 minus 119886 minus 119910 eacute

um caminho de comprimento 2 Assim podemos assumir que 119886 ne 119887 Se 119886119887 = 0 entatildeo 119909 minus 119886 minus

119887 minus 119910 eacute um caminho de comprimento 3 e portanto 119889 (119909 119910) le 3 Se 119886119887 ne 0 entatildeo 119886 minus 119886119887 minus 119910

eacute um caminho de comprimento 2 assim 119889 (119909 119910) = 2 Daiacute 119889 (119909 119910) le 3 e portanto

119889119894119886119898 (120591(119877)) le 3 ∎

Definiccedilatildeo 333 um anel diz-se artiniano se satisfaz a condiccedilatildeo de cadeia descendente ou

seja sobre ideais

31

Exemplos

Um domiacutenio de integridade artiniano eacute um corpo

Um anel com uma quantidade finita de ideais eacute artiniano Em particular um anel

finito (tal como 119885119899119885) eacute artiniano

Seja 119896 um corpo Entatildeo 119896[119905](119905119899) eacute artiniano para todo inteiro positivo 119899

Se 119868 eacute um ideal natildeo nulo de um domiacutenio de Dedekind 119860 entatildeo 119860119868 eacute um anel artiniano

de ideal principal

Teorema 334 Seja 119877 um anel artiniano comutativo (em particular 119877 poderia ser um anel

comutativo finito) Se 120591(119877) conteacutem um ciclo entatildeo 119892(120591(119877)) le 4 Em [36]

Demonstraccedilatildeo Suponha que 120591(119877) contenha um ciclo 119877 eacute um produto finito direto dos aneacuteis

locais artinianos Em primeiro lugar suponha que 119877 seja local com o ideal maximal diferente

de zero 119872 Entatildeo 119872 = 119886119899119899 (119909) para algum 119909 isin 119872lowast Se houver distintos 119910 119911 isin 119872lowast minus 119909 com

119910119911 = 0 entatildeo 119910 minus 119909 minus 119911 minus 119910 eacute um triacircngulo Caso contraacuterio 120591(119877) natildeo conteacutem ciclos uma

contradiccedilatildeo Neste caso 119892(120591(119877)) = 3 Em seguida suponha que 119877 = 1198771 times 1198772 Se ambos |1198771| ge

3 e |1198772| ge 3 entatildeo podemos escolher 119886119894 isin 119877119894 minus 0 1 Entatildeo (1 0) minus (0 1) minus (1198861 0) minus (0 1198862) minus

(1 0) eacute um quadrado Entatildeo neste caso 119892(120591(119877)) le 4 Assim podemos assumir que 1198771 = 1198852 Se

|119885(1198772)| le 2 entatildeo 120591(119877) natildeo conteacutem ciclos uma contradiccedilatildeo Portanto devemos ter |119885(1198772)| ge

3 Como o 120591(119877) eacute conexo existem distintos 119909 119910 isin 119885(1198772)lowast com 119909119910 = 0 Assim (0 119909 ) minus (1 119910) minus

(0 119910) minus (0 119909) eacute um triacircngulo Portanto neste caso 119892(120591(119877) ) = 3 Assim em todos os casos

119892(120591(119877)) le 4 ∎

Teorema 335 Seja R um anel comutativo Entatildeo 120591(119877) eacute completo se e somente se

119877 cong 1198852 times 1198852 119900119906 119909119910 = 0 119901119886119903119886 119905119900119889119900 119909 119910 isin 119885(119877) Em [13]

Demonstraccedilatildeo (larr) 119901119900119903 119889119890119891119894119899119894ccedilatilde119900

(rarr) Suponha que 120591(119877) esteja completo mas haacute um 119909 isin 119885(119877) com 1199092 ne 0 Mostramos que

1199092 = 119909 Se natildeo 1199093 = 1199092119909 = 0 Portanto 1199092 (119909 + 1199092) = 0 com 1199092 ne 0 entatildeo 119909 + 1199092 isin 119885(119877)

Se 119909 + 1199092 = 119909 entatildeo 1199092 = 0 uma contradiccedilatildeo Assim 119909 + 1199092 ne 119909 entatildeo 119909 2 = 119909 2 + 1199093 =

119909(119909 + 1199092) = 0 uma vez que 120591(119877) estaacute completa novamente uma contradiccedilatildeo Assim 1199092 = 119909

A prova do teorema anterior temos 119877 cong 1198852 times 119860 e necessariamente 119860 cong 1198852 ∎

Exemplo Para cada inteiro 119899 ge 1 seja 119877119899 = 1198852[119909](119909119899+1) um anel local finito Entatildeo 119909119899 eacute o

uacutenico veacutertice adjacente a cada outro veacutertice No entanto para 119899 ge 3 120591(119877119899) natildeo eacute um grafo

em estrela uma vez que os veacutertices 119909119899minus1 + 119909119899 119890 119909119899minus1 tambeacutem satildeo adjacentes Nota que

|120591(119877119899)| = 2119899 minus 1

32

Figura 23

34 Automorfismo de 120533(119825)

Para qualquer grafo 120591(119877) o grau de um veacutertice 119909 de 120591(119877) eacute 120575(119909) = |119910 isin 120591(119877)|119910 eacute adjacente a

119909 Para um veacutertice 119909 do grafo de divisores de zero 120591(119877) temos que 120575(119909) = |119886119899119899(119909) minus 0 119909|

Agora nos especializamos em 120591(119885119899) e estabelecemos alguma notaccedilatildeo Seja 119899 ge 4 natildeo eacute um

inteiro primo e seja 119883 = 119889 isin 119885|1 lt 119889 lt 119899 119890 119889|119899 Para cada 119889 isin 119883 seja 119881119889 = isin 119885119899|1 lt 119909 lt

119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|

Teorema 341 Seja 119899 ge 4 um inteiro natildeo-primo Entatildeo 119860119906119905(120591(119885119899)) eacute um produto (finito)

direto de grupos simeacutetricos Especificamente 119860119906119905(120591(119885119899)) cong 120561(119878119899119889119889 isin 119883) em que 119883 =

119889 isin 119885|1 lt 119889 lt 119899 119890 119889119899| e 119899119889 = |119909 isin 119885|1 lt 119909 lt 119899 119890 119898119889119888(119909 119899) = 119889(= |119881119889|) Em [13]

Demonstraccedilatildeo Use a mesma notaccedilatildeo como acima Como dois veacutertices de 120591(119885119899) tem o

mesmo grau se e somente se eles estiverem no mesmo 119881119889 e automorfismo do grafo preservam

o grau temos 119891(119881119889) = 119881119889 para cada 119891 isin 119860119906119905(120591(119885119899)) 119890 119889 isin 119883 Define 120593 119860119906119905(120591(119885119899)) rarr

120561119878119899119889|119889 isin 119883 119901119886119903119886 120593(119891) = (119891|119881119889) 119888119900119898 119891|119881119889 visto de forma natural como um elemento de 119878119899

Pelo comentaacuterio acima 120593 eacute um monomorfismo de grupo bem definido Para mostrar isso 120593 eacute

surjectivo basta mostrar que para cada 119889 isin 119883 fixo e a permutaccedilatildeo partir de 120572 119886 119881119889 existe

um 119891 isin 119860119906119905(120591(119885119899)) com 119891|119881119889 = 120572 119890 119891|119881119889 = 1119881119889 para todo 119889 ne 119889 119890119898 119883 Isso se segue pois para

qualquer 119909 119910 isin 119881119889 119890 119886 isin 119885119899 119886119909 = 0 se e somente se 119886119910 = 0 ∎

Corolaacuterio 342 Seja 119951 ge 120786 um nuacutemero inteiro natildeo primo Entatildeo

119860119906119905(120591(119885119899)) eacute trivial se e somente 119899 = 4

119860119906119905(120591(119885119899)) eacute abeliano se e somente 119904119890 119899 = 4 6 8 9 119900119906 12

Em particular 119860119906119905(120591(119885119899)) cong 1198852 quando 119899 = 6 8 119900119906 9 e 119860119906119905(120591(11988512)) cong 1198852 times 1198852 times 1198852

33

Exemplo Ilustramos a prova do Teorema 341 por computaccedilatildeo de 119860119906119905(120591(11988512)) Temos 119909 =

2 3 4 6 e assim 1198812 = 2 10 1198813 = 3 9 1198814 = 4 8 119890 1198816 = 6 Assim 119860119906119905(120591(11988512)) cong

1198782 times 1198782 times 1198782 times 1198781 cong 1198852 times 1198852 times 1198852 Isto tambeacutem eacute evidente a partir das simetrias oacutebvias do grafo

do divisor de zero de 11988512 abaixo

times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4

O GRAFO DOS DIVISORES DE ZERO DOS INTEIROS DE GAUSS MOacuteDULO 119899

Neste capiacutetulo estudamos o grafo dos divisores de zero de um anel de grande importacircncia o

anel dos inteiros de Gauss moacutedulo 119899

41 Introduccedilatildeo

Seja 119899 um nuacutemero natural e lang119899rang o ideal principal gerado por 119899 em 119885[119894] 119885119899 = 0 1 2 hellip 119899 minus 1 o

anel de inteiros modulo 119899 Entatildeo o anel de fator 119885[119894]lang119899rang eacute isomorfo para 119885119899[119894] = + 119887 isin

119885119899 o que implica que 119885119899[119894] eacute um anel de ideal principal O anel 119885119899[119894] eacute chamado de anel de

inteiros Gaussianos modulo 119899

Foi mostrado em Abu Osba [1] que + 119894 eacute uma unidade em 119885119899[119894] se e somente se 2 + 2 eacute

uma unidade em 119885119899 E se 119899 = prod 119886119895

119896119895119904119895=1 eacute a decomposiccedilatildeo prima do nuacutemero inteiro positivo 119899

entatildeo 119885119899[119894] eacute o produto direto dos aneacuteis 119885119886119895

119896119895 [119894] Tambeacutem se 119898 = 119905119896 para alguns primo 119905 e

inteiro positivo 119896 entatildeo 119885119898[119894] eacute local se e somente se 119905 = 2 119900119906 119905 equiv 3 (1198981199001198894)

Recordemos que em um grafo 120591(119877) Um conjunto dominante eacute um conjunto de veacutertices 119860 tal

que todo veacutertice fora de 119860 eacute adjacente a pelo menos um veacutertice em 119860 O nuacutemero dominante

de um grafo 120591(119877) denotado por 120575(120591) eacute o menor nuacutemero da forma |119860| onde 119860 eacute um conjunto

dominante

O caso quando 119899 eacute um primo ou a potecircncia de um primo eacute considerado primeiro Entatildeo o caso

geral eacute considerado

O nuacutemero de veacutertices em cada grafo o diacircmetro e a cintura satildeo encontrados

As caracterizaccedilotildees completas em termos de 119899 satildeo dadas nos casos em que o grafo 120591(119885119899[119894]) eacute

planar regular Euleriano completo ou bipartido completo

42 GRAFO DOS DIVISORES DE ZERO PARA 119833119853119847[119842]

Nesta seccedilatildeo as propriedades baacutesicas de 120591(119885119905119899[119894]) satildeo estudados Trecircs casos satildeo considerados

Quando 119905 = 2 119905 equiv 3 (1198981199001198894) 119900119906 119905 equiv 1 (1198981199001198894)

421 Grafo dos divisores de zero para Z2n[i]

Note-se que 2 natildeo eacute um primo Gaussiano uma vez que 2 = (1 + 119894)(1 minus 119894) no entanto 2 =

minus119894(1 + 119894)2 entatildeo 1198852[119894] eacute isomorfo para o anel local 119885[119894]lang(1 + 119894)2rang com seu uacutenico ideal

36

maximal 0 1 + 1198941 Observe que 1 minus 119894 = minus119894(1 + 119894) e entatildeo os dois elementos satildeo associados

em 119885[119894] e geram o mesmo ideal maximal Aleacutem disso em 1198852[119894] temos 1 + 1198941 = 1 minus 1198941 Assim

temos 119881(120591(1198852[119894])) = 1 + 1198941 o que implica que 120591(1198852[119894]) eacute o grafo nulo 1198731 isto eacute um grafo

com um veacutertice e sem arestas

Agora seja 119899 um inteiro superior a 1 Entatildeo 2119899 = (minus1)119899(1 + 119894)2119899 e assim

1198852119899[119894] cong119885[119894]

lang2119899rang= 119885[119894]lang(1 + 119894)2119899rang Daiacute 1198852119899[119894] eacute local com o seu uacutenico ideal maximal 119872 = lang1 + 1198941rang e

entatildeo 119881(120591(1198852119899[119894])) = lang1 + 1198941rang0 Eacute faacutecil provar o seguinte lema

Lema 4211 O uacutenico ideal maximal em 1198852119899[119894] eacute + 119894119887 119886 119890 119887 119904atilde119900 119886119898119887119900119904 119901119886119903119890119904 119900119906 iacute119898119901119886119903119890119904

Observe que (minus119894)119899minus1(1 + 1198941)2119899minus1 = (2)119899minus1(1 + 1198941) Por isso temos o seguinte Teorema

Teorema 4212 Seja 119899 gt 1 Entatildeo para todo 120572 isin 1198852119899[119894] 119905119890119898119900119904 120572(2)119899minus1(1 + 1198941) =

0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)

Demonstraccedilatildeo Se 120572 natildeo eacute uma unidade entatildeo 120572 = ( + 119894)(1 + 1198941) isin lang1 + 1198941rang o que implica

que 120572(2)119899minus1(1 + 1198941) = (2)119899minus1( + 119894)(1 + 1198941)2119899 = 0 Entatildeo suponha que 120572 eacute uma unidade e

portanto 120572 = + 119894 com 119886 119890 119887 natildeo satildeo nem pares nem iacutempares Assim (2)119899minus1(1 + 1198941)(120572 minus 1) =

(2)119899minus1(1 + 1198941)( minus 1 + 119894) = 0 jaacute que neste caso 119886 minus 1 119890 119887 satildeo ambos pares ou ambos satildeo

iacutempares e nesse caso 120572 minus 1 isin lang1 + 1198941rang Portanto 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941) ∎

Jaacute que 1198852119899[119894] eacute local com 119885(1198852119899[119894]) = lang1 + 1198941rang como seu ideal maximal 119885(1198852119899[119894]) eacute um

aniquilador ideal e portanto existe um veacutertice adjacente a cada veacutertice em 120591(1198852119899[119894])

Anderson e Livingston [7] Na verdade para qualquer veacutertice 120572 em 120591(1198852119899[119894]) 120572 eacute adjacente a

(1 + 1198941)2119899minus1

Teorema 4213 Para 119899 ge 1 |119881(120591(1198852119899[119894]))| = 22119899minus1 minus 1

Demonstraccedilatildeo O nuacutemero de unidades em 1198852119899[119894] eacute 22119899minus1 minus 1 119890119898 [30] Assim sendo

|119881(120591(1198852119899[119894]))| = 22119899 minus 22119899minus1 minus 1 = 22119899minus1 minus 1 ∎

Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2

Demonstraccedilatildeo 120591(1198852119899[119894]) natildeo eacute completo desde 2 119890 1 + 1198941 satildeo veacutertices em 120591(1198852119899[119894]) mas

2(1 + 1198941) ne 0 entatildeo para 120572(1 + 1198941)120573(1 + 1198941) tem veacutertices em 120591(1198852119899[119894]) com 120572 120573 isin

120591(1198852119899[119894]) 119890 120572(1 + 1198941)120573(1 + 1198941) ne 0 Entatildeo temos o caminho 120572(1 + 1198941) hellip (1 + 1198941)119899minus1 hellip 120573(1 +

1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37

Demonstraccedilatildeo Para 119899 = 2 temos o ciclo 2 ⋯ 1198942 ⋯ 2 + 1198942 ⋯ 2 Para 119899 gt 2 sempre temos os

ciclos 2119899minus1 ⋯ 2 ⋯ 1198942119899minus1 ⋯ 2119899minus1 portanto 119892(120591(1198852119899[119894])) = 3 ∎

Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])

422 Grafo dos Divisores de Zero para 119833119850119847[119842] 119850 equiv 120785(119846119848119837120786)

Se 119902 equiv 3(1198981199001198894) entatildeo 119902 eacute um primo Gaussiano e entatildeo 119885119902[119894] eacute um corpo de decomposiccedilatildeo

para o polinocircmio 119892(119909) = 1199092 + 1 sobre o corpo 119885119902 119890 119885119902[119894] eacute isomorfo para o corpo 119885[119894]lang119902rang

Entatildeo neste caso 119885119902[119894] natildeo tem divisores de zero diferente de zero

Se 119899 gt 1 entatildeo 119885119902119899[119894] cong 119885[119894]lang119902119899rang eacute anel local com ideal maximal lang119902rang Portanto

119881 (120591(119885119902119899[119894])) = langrang0

Para qualquer veacutertice 120572 em 120591(119885119902119899[119894]) 120572 eacute adjacente para 119899minus1 Na verdade neste caso se

120572 eacute um veacutertice em 120591(119885119902119899[119894]) entatildeo 120572 eacute adjacente a cada elemento em lang119899minus1rang0

Agora para determinar o nuacutemero de veacutertices em 120591(119885119902119899[119894])

Teorema 4221 Para 119899 gt 1 |119881 (120591(119885119902119899[119894]))| = 1199022119899minus2 minus 1

Demonstraccedilatildeo O nuacutemero de unidades em 119885119902119899[119894] eacute 1199022119899 minus 1199022119899minus2 Cross [20] Assim sendo

|119881 (120591(119885119902119899[119894]))| = |langrang| minus 1 = 1199022119899 minus (1199022119899 minus 1199022119899minus2) minus 1 = 1199022119899minus2 minus 1

38

Eacute claro que 120591(1198851199022[119894]) eacute um grafo completo 1198701199022minus1 e entatildeo 119889119894119886119898 (120591(119885119902119899[119894])) = 1 Para 119899 gt 2

119885119902119899[119894] natildeo eacute completo e 119889119894119886119898 (120591(119885119902119899[119894])) = 2 visto que se 120572 120573 satildeo veacutertices em

120591(119885119902119899[119894]) 120572 120573 isin 119885119902119899[119894] 119890 120572 120573 ne 0 entatildeo temos 120572 ne 119899minus1 e 120573 ne 119899minus1 Assim temos o

caminho 120572 hellip 119899minus1 hellip 120573 portanto o resultado ∎

Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3

Demonstraccedilatildeo Se 119899 = 2 119890119899119905atilde119900 120591(119885119902119899[119894]) eacute completo com mais de 3 veacutertices e assim

119892 (120591(119885119902119899[119894])) = 3 119878119890 119899 gt 2 sempre temos o ciclo 119899minus1 hellip hellip 119894119899minus1 hellip 119899minus1 portanto

119892 (120591(119885119902119899[119894])) = 3 ∎


Seja 119901 um inteiro primo que seja congruente com 1 moacutedulo 4 Entatildeo haacute um inteiro 119888 tal

que 1198882 equiv minus1 (119898119900119889119901) e existe tambeacutem um 119886 119887 isin 119885 de tal modo que 119901 = 1198862 + 1198872 = (119886 + 119894119887)(119886 minus

119894119887) aleacutem disso 119886 + 119894119887 119890 119886 minus 119894119887 satildeo primos Gaussianos em 119885[119894] Assim os ideais (119886 + 119894119887) 119890 (119886 minus

119894119887) satildeo os uacutenicos ideais maximais em 119885[119894] contendo 119901 jaacute que 119885[119894] eacute um domiacutenio de

factorizaccedilatildeo uacutenica Por isso temos 119885119901[119894] cong119885[119894]

lang119901rangcong (

119885[119894]

lang119886+119894119887rang) times (119885[119894]lang119886 minus 119894119887rang) Os ideais lang +

119894rang 119890 lang minus 119894rang satildeo os uacutenicos ideais maximais em 119885119901[119894] O nuacutemero de unidades em 119885119901[119894] eacute (119901 minus

1)2 Cross [20] o que implica que |119881 (120591(119885119901119899[119894]))| = 1199012 minus (119901 minus 1)2 minus 1 = 2119901 minus 2

Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39

Aleacutem disso 120591(119885119901[119894]) eacute o grafo bipartido completo 119870119901minus1 119901minus1 e portanto 119889119894119886119898 (120591(119885119901[119894])) = 2 e

119892 (120591(119885119901[119894])) = 4

Para investigar o caso mais geral para 119901 equiv 1 (1198981199001198894) 119899 gt 1 119890 119901 = 1198862 + 1198872 Entatildeo 119901119899 =

(1198862 + 1198872)119899 = (119886 + 119894119887)119899(119886 minus 119894119887)119899 e portanto 119901119899 estaacute contido apenas dois ideais maximais em

119885[119894] ou seja lang + 119894rang 119890 lang minus 119894rang Por isso temos 119885119901119899[119894] cong 119885[119894]lang119901119899rang cong (119885[119894]lang(119886 + 119894119887)119899rang) times

(119885[119894]lang(119886 minus 119894119887)119899rang)

Neste caso 119881 (120591(119885119901119899[119894])) = (lang + 119894rang cup lang minus 119894rang)0 O nuacutemero de unidades em 119885119901119899[119894] eacute (119901119899 minus

119901119899minus1)2 Cross [20] portanto temos o seguinte teorema

Teorema 4231 Para 119899 gt 1 119881 (120591(119885119901119899[119894])) = 21199012119899minus1 minus 1199012119899minus2 minus 1

Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3

Demonstraccedilatildeo Seja 119901 = 1198862 + 1198872 Eacute claro que 119889( + 119894 minus 119894) gt 1 Se existe + 119894 de modo

que ( minus 119894)( + 119894) = 0 = ( + 119894)( + 119894) entatildeo 119901119899 divide (119886119909 + 119887119910) (119886119910 minus 119887119909) (119886119909 minus

119887119910) 119890 (119886119910 + 119887119909) Entatildeo 119901119899 divide 2119886119909 119890 2119887119910 e portanto 119901119899 divide 119909 119890 119910 isto eacute + 119894 = 0

Assim 119889( + 119894 minus 119894) gt 2 Assim temos o caminho ( + 119894) hellip ( + 119894)119899minus1

( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1

hellip ( minus 119894) e visto que o diacircmetro de um grafo de divisor de zero de um anel

comutativo finito com unidade eacute sempre inferior ou igual a 3 em [7] obtemos o resultado ∎

Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3

Demonstraccedilatildeo Se 119899 = 2 entatildeo considere o ciclo hellip + 119894 hellip 119894 hellip Para 119899 gt 2 sempre

temos o ciclo ()119899minus1 hellip hellip 119894()119899minus1 hellip ()119899minus1 portanto 119892 (120591(119885119901119899[119894])) = 3 ∎

43 Grafo dos Divisores de Zero para 119833119847[119842]

Nesta seccedilatildeo os inteiros 119902 e 119902119895 satildeo usados implicitamente para denotar primos congruente

com 3 119898119900119889119906119897119900 4 enquanto 119901 119890 119901119904 denotam nuacutemeros inteiros congruentes com 1 119898oacute119889119906119897119900 4

O caso geral seraacute agora investigado Assumindo que 119899 = prod 119905119895119899119895119898

119895=1 A funccedilatildeo 120579 119885119899[119894] rarr

prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo

Seja agora 119899 = 2119896 times prod 119902119895120572119895119898

119895=1 times prod 119901119904120573119904119897

119904=1 Entatildeo o nuacutemero de unidades em 119885119899[119894] eacute 22119896minus1 times

prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 portanto temos o seguinte lema

40

Lema 431 Seja 119899 = 2119896 times prod 119902119895120572119895119898

119895=1 times prod 119901119904120573119904119897

119904=1 Entatildeo |119881 (120591(119885119901119899[119894]))| = 119899 minus (22119896minus1 times

prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1

Diacircmetro e Cintura para 120649(119937119953119951[119946])

Eacute mostrado em Axtell [10] que se 1198771 119890 1198772 satildeo aneacuteis comutativos com identidade e sem

divisores de zero diferente de zero entatildeo 119889119894119886119898(1198771 times 1198772) = 3 Usando isso juntamente com os

resultados acima obtemos o seguinte teorema

Teorema 432 Seja 119899 um inteiro positivo maior que 1 Entatildeo

1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022

2) 119889119894119886119898(120591(119885119899[119894])) = 2 se e somente se 119899 = 119901 119900119906 119899 = 2119898 119888119900119898 119898 ge 2 119900119906 119899 = 119902119898 119888119900119898 119898 ge 3

3) 119889119894119886119898(120591(119885119899[119894])) = 3 se e somente se 119898 = 119901119898 119888119900119898 119898 ge 2 119900119906 119899 eacute divisiacutevel pelo menos por

dois primos distintos

Foi mostrado anteriormente que para qualquer 119905 119890 119899 gt 1 119892(120591(119885119905119899[119894])) = 3 e 119892 (120591(119885119901[119894])) = 4

Agora estudamos mais casos

Teorema 433 Seja 119899 = prod 119905119895119899119895119898

119895=1 seja a factorizaccedilatildeo prima de 119899 Entatildeo

1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

3) Se 119899 = 1199011 times 1199012 119900119906 119899 = 1199011 times 119902 119900119906 119899 = 1199011 times 2 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

Demonstraccedilatildeo 1) suponha que 119899119896 gt 1 Definimos 119895 = 119905 119895 = 119896

0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898

119895=1 [119894] Entatildeo consideramos o ciclo 119899119896minus1 hellip 119894 hellip 119899119896minus1 + 119894119899119896minus1 hellip 119899119896minus1 portanto

119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1

119898 Entatildeo 119911 isin prod 119885119905119895

[119894]119898119895=1 e temos o ciclo hellip hellip 119911 hellip portanto 119892(120591(119885119899[119894])) = 3

3) Seja 1199011 = 1198862 + 1198872 Entatildeo temos ciclos

( + 119894 0) hellip (0 1) hellip ( minus 119894 0) hellip ( + 119894 0) portanto 119892(120591(119885119899[119894])) = 3

Para 4) e 5) Em Axtell [10] ∎

41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])

431 Quando 120533(119833119847[119842]) Eacute completo ou bipartido completo

Foi mostrado em Anderson e Livingston [7] que para um anel comutativo 119877 120591(119877) eacute completo

se e somente se 119877 cong 1198852 times 1198852 ou 119909119910 = 0 para todo o 119909 119910 isin 120591(119877)

Teorema 4311 O grafo 120591(119885119899[119894]) eacute completo se e somente se 119899 = 1199022

Demonstraccedilatildeo Foi mostrado anteriormente que se 119899 = 1199022 entatildeo 120591(119885119899[119894]) eacute um grafo

completo Assim Suponha que 120591(119885119899[119894]) eacute completo Entatildeo 119899 eacute um nuacutemero composto natildeo

divisiacutevel por 1198863 para qualquer nuacutemero primo 119886 uma vez que neste caso natildeo eacute adjacente a

119894 Aleacutem disso 119899 eacute natildeo eacute divisiacutevel por dois primos distintos 119886 119887 uma vez que neste caso eacute

um veacutertice em 120591(119885119899[119894]) mas natildeo eacute adjacente a 119894 Claramente 119901 ∤ 119899 pois se 119901 = 1198862 + 1198872

entatildeo natildeo eacute adjacente para + 119894 temos tambeacutem2 ∤ 119899 uma vez que 1 + 1198941 natildeo eacute

adjacente a 2 Entatildeo 119899 = 1199022 ∎

Eacute claro que se 120591(119877) eacute um grafo bipartido completo 119870119898119899 com 119898119894119899119898 119899 ge 2 entatildeo 119892(120591(119877)) =

4 entatildeo se 120591(119877) conteacutem um ciclo com comprimento 3 natildeo pode ser um grafo bipartido

completo ou mesmo bipartido

Lema 4332 Seja 119877 = 1198771 times 1198772 Entatildeo 120591(119877) eacute um grafo bipartido completo se e somente se

1198771119890 1198772 satildeo domiacutenios de integridade

Demonstraccedilatildeo Se 119877 = 1198771 times 1198772 onde 1198771119890 1198772 satildeo domiacutenios de integridades entatildeo 120591(119877) eacute um

grafo bipartido completo com 119860 = (119909 0) 119909 isin 11987710119890 119861 = (0 119887) 119887 isin 11987710 como o dois

42

conjuntos disjuntos de veacutertices de modo que todo veacutertice em 119860 seja adjacente a cada veacutertice

em 119861 e natildeo temos outra adjacecircncia Agora se 1198771 natildeo eacute um domiacutenio de integridade com

119909 119910 isin 11987710 119890 119909119910 = 0 entatildeo temos o 3-ciclo (119909 0) hellip (119910 0) hellip (0 1) hellip (119909 0) entatildeo 120591(119877) natildeo eacute

um grafo bipartido completo ∎

Se 119877 eacute um produto direto de mais de dois domiacutenios de integridades natildeo triviais entatildeo 119877 eacute

reduzido e a interseccedilatildeo de dois ideais principais natildeo eacute trivial entatildeo 120591(119877) eacute grafo bipartido

natildeo completo Akbari [2] Entatildeo se 119901 equiv 1 (1198981199001198894) com 119901 = 1198862 + 1198872 entatildeo 120591(119885119901[119894]) eacute um grafo

bipartido completo jaacute que 119885119901[119894] cong 119885[119894]lang119901rang cong 119885[119894]lang119886 + 119894119887rang times 119885[119894]lang119886 minus 119894119887rang E se 1199021 119890 1199022 satildeo dois

primos tais que 119902119895 equiv 3 (1198981199001198894) para cada 119895 entatildeo 120591(11988511990211199022[119894]) eacute um grafo bipartido completo

jaacute que 11988511990211199022[119894] cong 1198851199021

[119894] times 1198851199022[119894] um produto direto de dois corpos Eacute claro que 120591(1198854[119894]) natildeo eacute

um grafo bipartido completo de forma semelhante 120591(1198851199022[119894]) natildeo eacute um grafo bipartido

completo jaacute que eacute completo em mais de dois veacutertices 120591(1198851199012[119894]) natildeo eacute um grafo bipartido

completo uma vez que se 119901 = 1198862 + 1198872 entatildeo temos o 3-ciclo ( + 119894) hellip ( minus 119894) hellip 119894( +

119894) hellip ( + 119894) Se 119886 eacute um nuacutemero primo entatildeo 120591(1198851198863[119894]) natildeo eacute um grafo bipartido completo

uma vez que temos o 3-ciclo hellip 2 hellip 1198942 hellip Agora pode-se concluir facilmente com o

seguinte teorema

Teorema 4333 O grafo 120591(119885119899[119894]) eacute bipartido completo se e somente se 119899 = 119901 ou 119899 = 11990211199021

Em [7] Para um anel finito R se 120591(119877) eacute um grafo de estrelas entatildeo 119877 cong 1198852 times 119865 onde 119865 eacute um

corpo finito com |119865| ge 3 ou 119877 eacute local com ideal maximal 119872 satisfazendo 119877

119872cong 1198852 1198723 =

0 119890 |1198722| le 2

Teorema 4334 Para cada 119899 120591(119885119899[119894]) nunca eacute um grafo de estrelas

Demonstraccedilatildeo Assumindo 120591(119885119899[119894]) eacute um grafo de estrela Se 119885119899[119894] cong 1198852 times 119865 entatildeo 1198992 =

|119885119899[119894]| = |1198852 times 119865| = 2 times 119886119896 para um nuacutemero inteiro 119886 Portanto 119899 = 2119898 119898 gt 1 Se 119885119899[119894] eacute

local entatildeo qualquer 119899 = 2119898 119900119906 119899 = 119902119898 119898 gt 1 Abu Osba [1] aleacutem disso 2 119889119894119907119894119889119890 119899 entatildeo

novamente 119899 = 2119898 119898 gt 1 Mas neste caso temos o ciclo 2119898minus1 hellip 2 + 1198942 hellip 1198942119898minus1 hellip 2119898minus1 uma

contradiccedilatildeo Portanto 120591(119885119899[119894]) nunca eacute um grafo de estrelas ∎

432 Quando o Nuacutemero Dominante para 120533(119833119847[119842]) eacute 120783 119848119854 120784

Agora eacute caracterizada quando o nuacutemero dominante do grafo 120591(119885119899[119894]) eacute um ou dois

Teorema 4321 O nuacutemero dominante 120575(120591(119885119899[119894])) = 1 se e somente se 119899 = 2119898 119900119906 119899 = 119902119898

onde 119898 gt 1

Demonstraccedilatildeo Para 119898 gt 1 119904119890 119899 = 2119898 entatildeo cada elemento em 120591(119885119899[119894]) eacute adjacente a

(1 + 1198941)2119899minus1 119890 119904119890 119899 = 119902119898 entatildeo cada elemento eacute adjacente a 119898minus1 Se houver um veacutertice

43

adjacente a qualquer outro veacutertice entatildeo 119885119899[119894] eacute local ou 119885119899[119894] cong 1198852 times 119865 onde 119865 eacute um corpo

finito em [7] mas 119885119899 eacute local se e somente se 119899 = 2119898 119900119906 119899 = 119902119898 119890 119904119890 119885119899[119894] cong 1198852 times 119865 119890119899119905atilde119900 1198992 =

|119885119899[119894]| = |1198852 times 119865| = 2 times 119886119896 para algum nuacutemero primo 119886 portanto 119899 = 2119898 ∎

Teorema 4322 O nuacutemero dominante 120575(120591(119885119899[119894])) = 2 se e somente se 119899 = 119901119898 ou 119899 =

119886119898119887119896 119888119900119898 119886 119890 119887 satildeo primos distintos e natildeo congruentes com 1 119898oacute119889119906119897119900 4

Demonstraccedilatildeo Se 119899 = 119901119898 com 119901 = 1198862 + 1198872 entatildeo (119886 + 119894119887)119898 (119886 minus 119894119887)119898 eacute o menor

dominante o conjunto em 120591(119885119899[119894]) Se 119899 = 119886119898119887119896 entatildeo 120591(119885119886119898[119894]) conteacutem um veacutertice 120572 que eacute

adjacente a qualquer outro veacutertice em 120591(119885119886119898[119894]) 119890 120591(119885119887119896[119894]) conteacutem um veacutertice 120573 que eacute

adjacente a qualquer outro veacutertice em 120591(119885119887119896[119894]) Assim o conjunto (120572 0) (0 120573) eacute um

conjunto dominante em 120591(119885119886119898[119894]) times 120591(119885119887119896[119894]) asymp 120591(119885119899[119894]) de menor cardinalidade Portanto

120575(120591(119885119899[119894])) = 2

Assumindo 120575(120591(119885119899[119894])) = 2 119899 = prod 119886119895119899119895 119888119900119898 119896 ge 3119896

119895=1 e o conjunto dominante (119887119895)119895=1

119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896

Entatildeo 119887119895 = 0 119901119886119903119886 119905119900119889119900 119895 ne 119896 119890 119887119895 ne 0 Definimos

119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =

0 119901119886119903119886 119905119900119889119900 119895 ne 1 119890 1198881 ne 0 Agora definimos 119911119895 =

1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 uma contradiccedilatildeo

Assumindo que 119899 = 119901119898119888119896 119888119900119898 119901 = 119886 + 119894119887 e o conjunto dominante (120572 120573) (120574 120578) em 120591(119885119901119898 times

119885119888119896) Suponha que ( + 119894 1) times (120572 120573) = (0 0) Entatildeo ( + 119894)119898minus1

( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0

Assim 120578 ne 0 caso contraacuterio (1 0) times (120572 0) ne (0 0) e (1 0) times (120574 0) ne (0 0) Portanto ( minus

119894 1) times (120574 120578) ne (0 0) o que implica que ( minus 119894 1) times (120572 0) = (0 0) assim ( + 119894)119898

( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898

119889119894119907119894119889119890 120572 119894 119890 120572 = 0 uma contradiccedilatildeo

Assim = 119886119898119887119896 com 119886 119890 119887 satildeo primos natildeo congruentes com 1 119898oacute119889119906119897119900 4

Finalmente se 119899 eacute divisiacutevel por apenas um primo entatildeo 119899 = 119901119898 caso contraacuterio 120573(120591(119885119899)) = 1

∎

433 Quando 120533(119833119847[119842]) eacute planar

44

Proposiccedilatildeo 4331 O grafo eacute planar se e somente natildeo conteacutem subgrafo homeomoacuterfico para

1198705 119900119906 11987033

Teorema 4332 O grafo 120591(119885119899[119894]) eacute planar se e soacute se 119899 = 2 119900119906 4

Demonstraccedilatildeo Se 119899 eacute divisiacutevel por dois primos distintos 119886 119890 119887 entatildeo 119885119899[119894] tem um fator

direto da forma 119885119886119899[119894] times 119885119887119899[119894] e por isso contem um subgrafo homeomoacuterfico para 11987033 desde

|119885119899[119894]| = 1198992 ge 4 119901119886119903119886 119888119886119889119886 119899 gt 1 Entatildeo suponha que 119899 = 119886119898 para algum primo inteiro 119886 Se

119886 = 119901 = 1199092 + 1199102 entatildeo 119885119899[119894] cong (119885[119894]lang119909 + 119894119910rang) times (119885[119894]lang119909 minus 119894119910rang) com |(119885[119894]lang119909 + 119894119910rang)| =

|(119885[119894]lang119909 minus 119894119910rang)| = 119901119898 ge 5 assim 120591(119885119901119898[119894]) natildeo eacute planar Akbari [2] Se 119886 = 119902 entatildeo 119898 gt 1 e

|119885119902119898[119894]langrang| =1199022119898

1199022119898minus2 = 1199022 ge 9 e |119885119902119898[119894]| ge 92 = 81 assim 120591(119885119902119898[119894]) natildeo eacute planar Se 119886 =

2 119890 119898 = 1 entatildeo 120591(1198852[119894]) tem apenas um veacutertice e sem arestas o que mostra que 120591(1198852[119894]) eacute

planar Entatildeo suponha que 119898 gt 2 entatildeo |1198852119898[119894]| =22119898

22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])

natildeo eacute planar Akbari [2] Por isso acabamos com 119899 = 22 = 4 e neste caso o grafo 120591(1198854[119894]) eacute

planar ∎

434 Quando τ(Zn[i]) eacute regular

Agora estudamos quando o grafo 120591(119885119899[119894]) eacute regular Mas primeiro vamos provar o seguindo o

teorema

Teorema 4341 Seja 119905 um inteiro primo iacutempar 119899 gt 1 119890 119896 um nuacutemero inteiro

a) Se 1 le 119896 lt119899

2 119890119899119905atilde119900 120591(119885119905119899[119894]) 119905119890119898 119906119898 119907eacute119903119905119894119888119890 119889119890 119892119903119886119906 1199052119896 minus 1

b) Se 119899

2le 119896 lt 119899 119890119899119905atilde119900 120591(119885119905119899[119894]) 119905119890119898 119906119898 119907eacute119903119905119894119888119890 119889119890 119892119903119886119906 1199052119896 minus 2

Demonstraccedilatildeo Seja 119896 isin 1 2 3 hellip 119899 minus 1 e considere o veacutertice 119907 = (119905119899 minus 119905119896)(1 + 1198941)

Claramente 119905119899minus119896(1 + 1198941) isin 119873(119907) Para determinar 119873(119907) seja + 119894 ser um veacutertice em

120591(119885119905119899[119894]) com (0 0) ne ( ) ne (119905119899 minus 119905119896 119905119899 minus 119905119896 ) Entatildeo + 119894 isin 119873(119907) se e somente se ( +

119894)(119905119899 minus 119905119896)(1 + 1198941) = 0

(i) Se e somente se (119905119899 minus 119905119896)( minus ) = 0 = (119905119899 minus 119905119896)( + )

(ii) Se e somente se (119905119899 minus 119905119896)(119886 minus 119887) 119890 (119905119899 minus 119905119896)(119886 + 119887) satildeo muacuteltiplos de 119905119899

(iii) Se e somente se (119905119899minus119896 minus 1)(119886 minus 119887) 119890 (119905119899minus119896 minus 1)(119886 + 119887) satildeo muacuteltiplos de 119905119899minus119896

(iv) Se e somente se 119905119899minus119896 divide ambos 119886 minus 119887 119890 119886 + 119887 visto que 119905119899minus119896 119890 119905119899minus119896 minus 1 satildeo

relativamente primos

(v) Se e somente se 119905119899minus119896 divide ambos 2119886 119890 2119887

(vi) Se e somente se 119905119899minus119896 divide ambos 119886 119890 119887 119889119890119904119889119890 119905 eacute 119906119898 119901119903119894119898119900 iacute119898119901119886119903

45

Isso implica que 119873(119907) = 119878(0 119907) onde 119878 = (119905119899minus119896(119888 + 119894)) 119888 119889 isin 123 hellip 119905119896 minus 1

Claramente 0 isin 119878 119898119886119904 119907 natildeo precisa De fato se 119907 isin 119878 entatildeo 119888119905119899minus119896 = 119889119905119899minus119896 = 119905119899 minus 119905119896 o que

implica que 119888 = 119905119896 minus 1199052119896minus119899 e portanto 2119896 minus 119899 ge 0 119894 119890 119896 ge119899

2

Assim sendo para 1 le 119896 lt119899

2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899

2le 119896 lt 119899 temos 0 le 2119896 minus 119899 lt

119899 119890 119905119890119898119900119904 119907 = (119905119896 minus 1199052119896minus119899)(119905119899minus119896 + 119894119905119899minus119896) isin 119878 Isso implica que

119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎

Corolaacuterio 4342 Seja 119905 um inteiro primo iacutempar e 119899 gt 2 Entatildeo 120591(119885119905119899[119894]) natildeo eacute regular

Para qualquer inteiro primo iacutempar 119902 tal que 119902 equiv 3(1198981199001198894) 120591(119885119902[119894]) eacute vazio enquanto

120591(1198851199022[119894]) eacute completo Para qualquer inteiro primo iacutempar 119901 tal que 119901 equiv 1(1198981199001198894) e 119901 = 1198862 + 1198872

120591(119885119901[119894]) eacute um grafo bipartido completo enquanto 120591(1198851199012[119894]) natildeo eacute regular jaacute que 119873 =

(( + 119894)) = lang minus 119894rang0 mas 119873( + 119894) = lang( minus 119894)0rang 119873nesub (119901(119886 + 119894119887))

120591(1198852[119894]) natildeo tem arestas 120591(1198854[119894]) natildeo eacute regular e para qualquer 119899 gt 2 120591(1198852119899[119894]) eacute natildeo

regular uma vez que (1 + 1198941)2119899minus1 eacute adjacente a qualquer outro veacutertice enquanto 1 + 1198941 natildeo

eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()

Portanto 120591(119885119899[119894]) natildeo eacute regular Entatildeo temos o seguinte teorema

Teorema 4343 O grafo 120591(119885119899[119894]) eacute regular se e somente se 119899 = 2 119900119906 119899 = 119901 119900119906 119899 = 1199022

435 Quando 120533(119833119847[119842]) eacute Euleriano

Agora eacute o momento de caracterizar em termos de 119899 os casos em que o grafo 120591(119885119899[119894])

eacute Euleriano mas primeiro lembramos a seguinte proposiccedilatildeo bem conhecida

Proposiccedilatildeo 4351 O grafo conexo 120591(119877) eacute Euleriano se e somente se grau de cada veacutertice de

120591(119877) eacute par

Segue-se pelo Teorema 4341 acima que para qualquer primo iacutempar 119905 e 119899 gt 1 o grafo

120591(119885119905119899)[119894] conteacutem um veacutertice de graus iacutempar = 1199052119896 minus 2 e natildeo eacute Euleriano Se 119899 = 1 120591(119885119901[119894]) eacute

o grafo bipartido completo 119870119901minus1119901minus1 e tambeacutem eacute Euleriano Para 119905 = 2 eacute claro que 120591(1198852[119894]) eacute

Euleriano Para 119899 gt 1 O seguinte lema mostra que 120591(1198852119899[119894]) natildeo poderia ser Euleriano

46

Lema 4352 Para 119899 gt 1 o grafo 120591(1198852119899[119894]) tem um veacutertice de grau 1

Demonstraccedilatildeo Suponhamos que (1 + 1198941)( + 119894) = 0 Entatildeo 2119899 119889119894119907119894119889119890 (119909 minus 119910) 119890 (119909 + 119910) e

entatildeo 2119899 119889119894119907119894119889119890 2119909 119890 2119910 Portanto 119909 = 2119899minus1119886 119890 119910 = 2119899minus1119887 Entatildeo para obter uma soluccedilatildeo

diferente de zero para + 119894 devemos ter 119886 = 2119898 + 1 119890 119887 = 2119896 + 1 Mas neste caso + 119894 =

2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎

Reunindo os resultados acima obtemos o seguinte Teorema

Teorema 4353 Para um primo inteiro 119905 119890 119899 ge 1 O grafo 120591(119885119905119899)[119894] eacute Euleriano se e somente

se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)

Para o caso geral note primeiro que se 119899 gt 1 e (119909119895)119895=1

119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =

prod 119860119899119899(119909119895)119899119895=1 e visto que 119889119890119892(119909119895) = |119860119899119899(119909119895)| minus 1 segue que 119889119890119892 ((119909119895)

119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus

1 = |prod 119860119899119899(119909119895)119899119895=1 | minus 1 portanto 119889119890119892 ((119909119895)

119895=1

119899) eacute mesmo se e somente se |119860119899119899(119909119895)| eacute iacutempar

para todo 119895 Assim 120591(119877) eacute Euleriano se e somente se |119877119895| eacute iacutempar para todo 119895 119890 119904119890 119877119895 natildeo eacute um

domiacutenio de integridade 120591(119877119895) eacute Euleriano

Teorema 4354 O grafo 120591(119885119899[119894]) eacute Euleriano se e somente se 119899 = 2 119900119906 119899 eacute um primo

congruente com 1 119898oacute119889119906119897119900 4 119900119906 119899 eacute um inteiro composto que eacute um produto distinto de primos

436 Quando 120533(119833119847[119842]) eacute local 119815

Um grafo em que todos os veacutertices tem o mesmo grau eacute chamado de grafo regular Se todos

os veacutertices em um grafo 119866 tiverem vizinhanccedila que sejam isomoacuterficos para o mesmo grafo 119867

entatildeo 119866 eacute dito ser localmente 119867 Um grafo 119866 de diacircmetro 119889 eacute chamado de distacircncia regular

com paracircmetros 119901119894119895119896 0 le 119894 119895 119896 le 119889 se para cada triplo (119894 119895 119896) e para qualquer par (119906 119907) de

veacutertices de 119866 tal que 119889(119906 119907) = 119896 o nuacutemero de veacutertices na distacircncia 119894 de 119906 e a distacircncia 119895 de 119907

eacute 119901119894119895119896 cada um desses nuacutemeros 119901119894119895

119896 eacute independente da escolha particular de veacutertices Uma

classe especial de grafos regulares de distacircncia eacute a dos grafos fortemente regulares Um grafo

119866 eacute chamado fortemente regular se for a distacircncia regular do diacircmetro 2

Nesta seccedilatildeo investigamos os casos em que o grafo 120591(119885119899[119894]) eacute localmente 119867

Teorema 4361 O grafo 120591(119885119899[119894]) eacute localmente 119867 se e somente se 119899 = 2 119900119906 119899 = 119901 119900119906 119899 = 1199022

Demonstraccedilatildeo O grafo 120591(1198852[119894]) conteacutem apenas um veacutertice ou seja 1 + 119894 e assim 120591(1198852[119894]) eacute

localmente empty

47

Se 119899 = 119901 entatildeo 119899 = 1198862 + 1198872 para alguns 119886 119887 isin 119873 e o conjunto de veacutertices de 120591(119885119899[119894]) eacute

(lang + 119894rang cup lang minus 119894rang) minus 0 Nesse caso 120591(119885119899[119894]) eacute o grafo bipartido completo 119870119899minus1119899minus1 Daiacute o

grafo 120591(119885119899[119894]) eacute localmente (119899 minus 1) 1198701

Se 119899 = 1199022 o conjunto de veacutertices de 120591(119885119899[119894]) eacute langrang minus 0 Neste caso 120591(119885119899[119894]) eacute um grafo

completo 119870119899minus1 Portanto o grafo 120591(119885119899[119894]) eacute localmente 119870119899minus2

Foi mostrado em [1] que o grafo 120591(119885119899[119894]) eacute regular se e somente se 119899 = 2 119900119906 119899 = 119901 119900119906 119899 = 1199022

Por isso 120591(119885119899[119894]) natildeo pode ser local 119867 para qualquer outro caso Como o caso regular ∎

Uma vez que o grafo bipartido completo regular 119870119899119899 119899 ge 2 eacute fortemente regular e o grafo

completo 119870119899 eacute uma distacircncia regular pode-se deduzir o seguinte corolaacuterio

Corolaacuterio 4362 (a) O grafo 120591(119885119899[119894]) eacute localmente 119867 se e somente se for regular a distacircncia

se e somente se for regular

(b) O grafo 120591(119885119899[119894]) eacute fortemente regular se e somente se 119899 = 119901

437 Quando 120533(119833119847[119842]) eacute Halmitoniano

Um componente de um grafo natildeo dirigido eacute um subgrafo em que qualquer dois veacutertices satildeo

conexos uns aos outros por caminhos e ao qual natildeo podem ser adicionados mais veacutertices ou

arestas preservando sua conexidade ou seja eacute um subgrafo subordinado maacuteximo Para um

grafo 119866 seja 119888(119866) indicar o nuacutemero de componentes Um ciclo Hamiltoniano de um grafo 119866 eacute

um ciclo que conteacutem todos os veacutertices de 119866 Um grafo eacute Hamiltoniano se ele conteacutem um ciclo

Hamiltoniano

O nome ciclo Hamiltoniano decorre do fato de que Sir William Hamilton investigou sua

existecircncia no grafo do dodecaedro Um dos principais problemas natildeo resolvidos da teoria dos

grafos satildeo a obtenccedilatildeo de caracterizaccedilotildees simples para os grafos Hamiltonianos A maioria dos

teoremas existentes tem a forma se 119866 tiver limites suficientes entatildeo 119866 eacute Hamiltoniano

Provavelmente o mais famoso deles eacute o seguinte resultado

Proposiccedilatildeo 4371 Se 119866 eacute um grafo com 119899(ge 3) veacutertices e se 119889119890119892(119907) ge119899

2 para cada veacutertice

119907 entatildeo 119866 eacute Hamiltoniano

Proposiccedilatildeo 4372 Se 119866 eacute um grafo Hamiltoniano e 119878 eacute qualquer subconjunto proacuteprio natildeo

vazio de veacutertices em 119866 entatildeo 119888(119866 minus 119878) le |119878|

Vamos usar essas duas proposiccedilotildees para caracterizar quando o grafo 120591(119885119899[119894]) eacute Hamiltoniano

Vamos mostrar que 120591(119885119899[119894]) eacute Hamiltoniano se e somente se 119899 = 119901 119900119906 119899 = 1199022

Teorema 4373 Para cada 119898 ge 1 o grafo 120591(1198852119898[119894]) natildeo eacute Hamiltoniano

48

Demonstraccedilatildeo O grafo 120591(1198852[119894]) eacute o grafo trivial 1198701 que natildeo eacute Hamiltoniano Para 119898 gt 1 o

veacutertice define 119881(120591(1198852119898[119894])) = lang1 + 119894rang ndash 0 e neste grafo (1 + 119894)(1 minus 119894) = 2 ne 0 e todos os

veacutertices satildeo adjacentes a (1 + 119894)2119898minus1 Tambeacutem deg(1 + 119894) = 1 = deg(1 minus 119894) Em [1] Seja 119878 =

(1 + 119894)2119898minus1 e seja 119867 = 1 + 119894 1 minus 119894 Entatildeo 119888(120591(1198852119898[119894])) minus 119878) ge |119867| = 2 gt 1 = |119878| Portanto

segue pela Proposiccedilatildeo 4372 que 120591(1198852119898[119894]) natildeo eacute Hamiltoniano ∎

Teorema 4374 O grafo 120591(119885119901119898[119894]) eacute Hamiltoniano se e somente se 119898 = 1

Demonstraccedilatildeo Seja 119901 = 1198862 + 1198872 para algum 119886 119887 isin 119873 120591(119885119901[119894]) eacute um grafo bipartido completo

119870119901minus1119901minus1 com os dois conjuntos de veacutertices 1198811 = lang + 119894rang minus 0 e 1198812 = lang + 119894rang minus 0 Portanto eacute

claro que 120591(119885119901119898[119894]) eacute um grafo Hamiltoniano Agora seja 119898 gt 1 119885119901119898[119894] ≃ 119885119901119898 times 119885119901119898 entatildeo

seja 119878 = (0 120572119901119898minus1) isin 119885119901119898 times 119885119901119898 119898119889119888 (120572 119901) = 1 1198671 = (1 120572119901) isin 119885119901119898 times 119885119901119898 ∶ 119898119889119888(120572 119901) = 1 e

1198672 = (2 120572119901) isin 119885119901119898 times 119885119901119898 ∶ 119898119889119888(120572 119901) = 1 Entatildeo |1198671| = |1198672| ge 119901 minus 1 = |119878| Elementos de

1198671 119890 1198672 satildeo adjacentes apenas aos elementos de 119878 Entatildeo 119888(120591(119885119901119898 times 119885119901119898) minus 119878) ge |1198671| + |1198672| gt

|119878 | Por isso 120591(119885119901119898[119894]) natildeo eacute hamiltoniano ∎

Lema 4375 Seja 119898 gt 1 e seja 120572 120573 isin 0 119902 2119902 3119902 hellip (119902 minus 1)119902 sube 119885119902119898[119894] de tal modo que

(120572 120573) ne (0 0) Entatildeo o conjunto + 119894 ( + 119894)( + 119894) = 0 + 119894 ne 0 = lang119898minus1rang minus 0

Demonstraccedilatildeo Suponha que (119886119902 + 119887119902 119894)( + 119894) = 0 onde 119886 119887 isin 0 1 2 hellip 119902 minus 1 mas nem

ambos satildeo zero Entatildeo temos

119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972

Assim (1198862 + 1198872)119909 = 119902119898minus1(1198861198971 + 1198861198972) e (1198862 + 1198872)119910 = 119902119898minus1(1198861198972 minus 1198871198971) isso implica que 119902119898minus1|119909 e

119902119898minus1|119910 porque se 119902|(1198862 + 1198872) entatildeo (119886minus1119887)2 equiv minus1(119898119900119889119902) uma contradiccedilatildeo de fato que 119902 equiv

3(1198981199001198894) Portanto + 119894 isin lang119898minus1rang minus 0 ∎


Demonstraccedilatildeo 119885119902[119894] eacute um corpo e assim 120591(119885119902[119894]) eacute um grafo vazio 120591(1198851199022[119894]) eacute o grafo

completo 1198701199022minus1 em [1] que eacute um grafo Hamiltoniano Agora seja 119898 gt 2 Entatildeo o conjunto de

veacutertices de 120591(119885119902119898[119894] eacute langrang minus 0) Seja 119878 = lang119898minus1rang0 e seja 119867 = + 119894 120572 120573 isin

0 119902 2119902 3119902 (119902 minus 1) 119902 (120572 120573) ne (0 0) Entatildeo 119867 sube 119881(120591(119885119902119898[119894])) minus 119878 e segue pelo Lema

4375 que 119888(120591(119885119902119898[119894]) minus 119878) gt |119867| = 1199022 minus 1 = |119878| Assim segue pela Proposiccedilatildeo 4372 que

120591(119885119902119898[119894]) natildeo eacute Hamiltoniano ∎

Lema 4377 Se 119877 = 1198771 times 1198772 119888119900119898 |119903119890119892(1198771)| gt 1 e |119885lowast(1198772)| gt 1 entatildeo 120591(119877) natildeo eacute

Hamiltoniano

49

Demonstraccedilatildeo Seja 119878 = (0 119907) 119907 isin 119885lowast(1198772) e seja 119867 = (119906 119907) 119906 isin 119903119890119892(1198771) 119890 119907 isin 119885lowast(1198772)

Entatildeo os elementos de 119867 satildeo adjacentes apenas aos elementos de 119878 e 119888(120591(119877) minus 119878) ge |119867| =

|119903119890119892(1198771)| times |119885lowast(1198772)| ge 2|119885lowast(1198772) | gt |119885lowast(1198772) | = |119878| Assim 120591(119877) natildeo eacute Hamiltoniano

∎

Teorema 4378 Se um nuacutemero inteiro 119899 eacute divisiacutevel por pelo menos dois primos distintos

entatildeo 120591(119885119899[119894]) natildeo eacute Hamiltoniano

Demonstraccedilatildeo Se 119899 = 2119905 com 119898119889119888(2 119905) = 1 entatildeo 119885119899[119894] ≃ 1198852[119894] times 119885119905[119894] Seja 119878 = (1 +

119894 0) 119890 119867 = (1 + 119894 119907) 119907 isin 119880(119885119905[119894]) Entatildeo os veacutertices de 119867 satildeo adjacentes apenas a (1 +

119894 0) e portanto 119888(120591(1198852[119894] times 119885119905[119894]) minus 119878) ge |119867| gt 1 = |119878| entatildeo 120591(1198852119905[119894]) natildeo eacute Hamiltoniano

Para os outros casos se 119899 = 119898119896 com 119898 119896 gt 2 e 119898119889119888(119898 119896) = 1 entatildeo 119885119899[119894] ≃ 119885119898[119894] times 119885119896[119894] Se

nem 119885119898[119894] nem 119885119896[119894] eacute um corpo entatildeo o resultado segue imediatamente a partir do Lema

4377 Entatildeo suponha que ambos 119885119898[119894] e 119885119896[119894] satildeo corpos com 119898 lt 119896 seja 119867 = (0 119907) 119907 isin

(119885119898[119894] )lowast e seja 119878 = (119906 0) 119906 isin (119885119898[119894] )lowast Entatildeo os elementos de 119867 satildeo adjacentes apenas aos

elementos de 119878 e 119888(120591(119885119899[119894]) minus 119878) = |119867| = 119896 2 minus 1 gt 1198982 minus 1 = |119878| Assim 120591(119885119899[119894]) natildeo eacute

Hamiltoniano ∎

Combinando esses resultados em grafos Hamiltonianos juntamente com o Teorema 4361 e

Corolaacuterio 4362 podemos obter

Corolaacuterio 4379 Para 119899 gt 2 os seguintes satildeo equivalentes

1) 120591(119885119899[119894]) eacute Hamiltoniano

2) 120591(119885119899[119894]) eacute localmente 119867

3) 120591(119885119899[119894]) eacute regular

4) 120591(119885119899[119894]) eacute a distacircncia regular

5) 119899 = 119901 119900119906 119899 = 1199022

Teorema 43710 Para qualquer nuacutemero inteiro 119899 gt 1

1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1

Foi mostrado em Anderson e Livingston [1 23] que para um anel comutativo 119877 o grafo 120591(119877)

eacute conexo e tem diacircmetro no maacuteximo de 3 Portanto em vista do Teorema 43710 se

119899 ne 2119898 ou 119902119898 entatildeo 119903119886119889(120591(119885119899[119894]) isin 2 3 Agora consideramos o caso 119899 = 119901119898

Teorema 43711 Para qualquer nuacutemero inteiro 119898 ge 1 119903119886119889 (120591(119885119901119898[119894])) = 2

Demonstraccedilatildeo Seja 119901 = 1198862 + 1198872 Conforme mostrado em [8 Teorema 20] o conjunto

( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898

eacute um conjunto dominante miacutenimo de

50

120591(119885119901119898[119894]) Portanto 119903119886119889(120591(119885119901119898[119894]) gt 1 Sendo ( + 119894)119898

( minus 119894)119898minus1

eacute adjacente com ( +

119894)119898minus1

( minus 119894)119898

temos para qualquer veacutertice 120572 de 120591(119885119901119898[119894]) que natildeo eacute adjacente a ( +

119894)119898

( minus 119894)119898minus1

o veacutertice ( + 119894)119898minus1

( minus 119894)119898

eacute um vizinho comum de ( + 119894)119898

( minus 119894)119898minus1

e 120572 Portanto o veacutertice ( + 119894)119898

( minus 119894)119898minus1

tem excentricidade 2 e portanto

119903119886119889 (120591(119885119901119898[119894])) = 2 ∎

O resultado a seguir determina o raio para o caso restante em que 119899 possui pelo menos dois

fatores primos distintos

Teorema 43712 Seja 119899 um nuacutemero inteiro positivo com pelo menos dois fatores primos

distintos Entatildeo 119903119886119889(120591(119885119899[119894]) = 2

Demonstraccedilatildeo Seja 119899 = 119905119898119896 onde 119905 eacute um nuacutemero primo e 119898119889119888(119905 119896) = 1 Pelo Teorema

43710 119903119886119889(120591(119885119899[119894]) gt 1 Entatildeo seria suficiente encontrar um veacutertice em 120591(119885119899[119894]) com

excentricidade 2 Temos 120591(119885119899[119894]) ≃ 120591(119885119905119898[119894] times 119885119896[119894]) Observe que o conjunto de veacutertices de

120591(119885119905119898[119894] times 119885119896[119894]) eacute 1198601 cup 1198602 cup 1198603 cup 1198604 119900119899119889119890

1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

1198603 = (119909 119911) 119909 isin 119885119905119898[119894] minus 0 119885lowast(119885119896[119894]) e

1198604 = (119911 119910) 119911 isin 119885lowast(119885119905119898[119894]) 119910 isin 119885119896[119894] minus 0

onde 1198603 eacute vazio quando 119896 = 1199021 para alguns 1199021 e 1198604 eacute vazio quando 119905119898 = 1199022 para alguns 1199022

Considere o veacutertice 119907 = (119886 0) onde 119886 eacute um veacutertice de 120591(119885119905119898[119894]) com o miacutenimo

excentricidade Mostraremos que 119907 tem excentricidade 2 em 120591(119885119905119898[119894] times 119885119896[119894]) Como cada

veacutertice em 1198601 eacute adjacente a cada veacutertice em 1198602 temos 119889(119907 120572) le 2 para cada 120572 isin 1198601 cup 1198602 Se

(119909 119911) isin 1198603 existe um elemento 1199111 isin 119885lowast(119885119896[119894]) 119905119886119897 119902119906119890 1199111199111 = 0 e portanto (0 1199111) eacute um vizinho

comum de (119886 0) e (119909 119911) Assim 119889(119907 (119909 119911)) le 2 Finalmente se (119911 119910) isin 1198604 entatildeo pela escolha

de 119886 e de acordo com o Teorema 43710 ou o Teorema 43711 temos 119889(119886 119911) le 2 Entatildeo

qualquer 119911 = 119886 ou 119886_119911 isin 119864(120591(119885119905119898[119894])) ou 119886 119890 119911 tecircm um comum vizinho 1199111 em 120591(119885119905119898[119894])

Portanto se 119911 = 119886 entatildeo (119886 0) eacute adjacente a (119911 119910) ou o veacutertice (1199111 0) eacute um vizinho comum

de (119886 0) 119890 (119911 119910) e portanto em qualquer caso temos 119889(119907 (119911 119910)) le 2 Entatildeo suponha que 119911 =

119886 Agora se 119905119898 = 2 entatildeo 119886 = 1 + 1119894 = 119911 119890 (119886 0) eacute adjacente a (119911 119910) o que implica que

119889(119907 (119911 119910)) = 1 Se 119905119898 ne 2 entatildeo 119886 tem um vizinho 1199091 em 120591(119885119905119898[119894]) e portanto (1199091 0) eacute um

vizinho comum de (119886 0) 119890 (119911 119910) o que implica que 119889(119907 (119911 119910)) le 2 Portanto o veacutertice 119907 tem

excentricidade no maacuteximo 2 e portanto sua excentricidade eacute 2 Assim 119903119886119889(120591(119885119899[119894]) = 2

∎

51

Resumindo os resultados nos trecircs teoremas desta seccedilatildeo temos para quaisquer inteiros 119899 gt

1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900

44 O Grafo de Linha do Grafo de Divisor de Zero para o Anel de Inteiros

Gaussianos Modulo 119847

Definiccedilatildeo 441 O grafo de Linha eacute denotado por 119871(119866) e representa a adjacecircncia entre as

arestas do grafo 119866

Cada veacutertice de 119871(119866) representa uma aresta em 119866

Dois veacutertices de 119871(119866) satildeo adjacentes se e somente suas arestas correspondentes

compartilham um mesmo veacutertice em 119866 ou seja satildeo adjacentes em 119866

442 Quando 119819(120533(119833119847[119842])) eacute Euleriano

Agora eacute caracterizada quando o grafo de linha 119871(120591(119885119899[119894])) eacute Euleriano Antes de prosseguir

provamos o seguinte Lema

Lema 4421 (i) Cada veacutertice de 120591(119885119899[119894]) tem grau par se e somente se 119899 = 2 119901 119900119906 119899 eacute um

inteiro composto que eacute um produto de primos iacutempares distintos

(ii) Se 119899 = 119905119898 119898 gt 2 119890 119899 ne 1199022 entatildeo 120591(119885119899[119894]) tem um veacutertice de grau impar e outro de grau

par

(iii) Cada veacutertice de 120591(119885119899[119894]) tem grau iacutempar se e somente se 119899 = 1199022

Demonstraccedilatildeo (i) Como o grafo 119866 eacute Euleriano se e somente se cada veacutertice tiver um grau

par [1]

(ii) suponha que 119899 = 119905119898 119905 eacute 119901119903119894119898119900 119898 ge 2 119890 119899 ne 1199022 Entatildeo temos trecircs casos

Caso 1 (119905 = 2) entatildeo 119889119890119892(1 + 119894) = 1 e 119889119890119892(2119898minus1 + 2119898minus1119894) = 22119898minus1 minus 2

Caso 2 (119905 eacute 119906119898 119901119903119894119898119900 iacute119898119901119886119903 119890 119898 gt 2) Pelo Teorema 23 [120783] 120591(119885119899[119894]) tem um veacutertice de

grau 1199052119896minus1 minus 1 119900119899119889119890 1 le 119896 lt 1198982 e um veacutertice de grau 1199052119896 minus 2 119900119899119889119890119898

2le 119896 lt 119898

Caso 3 (119905 = 119901 = 1198862 + 1198872 119890 119898 = 2) Jaacute que 119889119890119892(119886 + 119894119887) = |119901lang119886 minus 119894119887rang| minus 1 e |119901lang119886 minus 119894119887rang| divide

|1198851199012| |119901lang119886 minus 119894119887rang| eacute iacutempar e portanto 119889119890119892(119886 + 119894119887) eacute mesmo

52

(iii) (rarr) Seja 119899 = prod 119886119895119898119895119896

119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886

Agora se todo 119886119895acute119904 satildeo primos iacutempares entatildeo 119889119890119892(119895) =

119899

(119886119895119898)

minus 1 e se 1198861 = 2 entatildeo 119889119890119892(1199091) =

(11989921198981) minus 1

(larr) Observe que 120591(1198851199022[119894]) cong 1198701199022minus1 Assim 119889119890119892(119907) = 1199022 minus 2 para cada veacutertice 119907 em 120591(1198851199022[119894])

Teorema 4422 (i) 119871(120591(119885119899[119894])) eacute grafo Euleriano se e somente se 119899 = 2 119901 1199022 119900119906 119899 eacute um

inteiro composto que eacute um produto de primos iacutempares distintos Em [16]

(ii) 119871(120591(119885119899[119894])) o grafo Euleriano natildeo implica necessariamente que 120591(119885119899[119894]) eacute

Euleriano

443 Quando 119819(120533(119833119847[119842])) eacute Hamiltoniano ou Planar

Teorema 4431 (i) se 119866 eacute um grafo de diacircmetro no maacuteximo de 2 com |119881(119866)| ge 4 entatildeo

119871(119866) eacute Hamiltoniano

(ii) O grafo de linha de um grafo Euleriano eacute Hamiltoniano e Euleriano

Se 119899 = 119901 2119898 119900119906 119902119898 119900119899119889119890 119898 ge 2 119890119899119905atilde119900 119889119894119886119898(120591(119885119899[119894])) le 2 Por outro lado se 119899 = 2 119901 119900119906 119899 eacute

um inteiro iacutempar composto que eacute um produto de primos distintos entatildeo 120591(119885119899[119894]) eacute Euleriano

Assim obteacutem-se o seguinte corolaacuterio

Corolaacuterio 4432 (i) Se 119899 = 119901 2119898 119900119906 119902119898 119900119899119889119890 119898 ge 2 entatildeo 119871(120591(119885119899[119894])) eacute Hamiltoniano

(ii) Se 119899 eacute um inteiro composto iacutempar que eacute um produto de primos distintos entatildeo

119871(120591(119885119899[119894])) eacute Hamiltoniano e Euleriano

Teorema 4433 Um grafo natildeo vazio 119866 tem um grafo de linha do planar 119871(119866) se e somente

se

(i) 119866 eacute planar

(ii) ∆(119866) le 4

(iii) Se 119889119890119892119866(119907) = 4 119890119899119905atilde119900 119907 eacute 119906119898 119907eacute119903119905119894119888119890 119888119900119903119905119886119889119900

Lembre-se de que 120591(119885119899[119894]) eacute planar se e somente se 119899 = 2 119900119906 119899 = 4 Mas 119871(120591(1198854[119894])) natildeo eacute

planar visto que ∆(120591(1198854[119894])) = 7 gt 4 Portanto obtemos o seguinte teorema

53

Teorema 4434 o grafo 119871(120591(119885119899[119894])) nunca eacute planar

444 Os nuacutemeros Cromaacutetico e Clique de 119819(120533(119833119847[119842]))

Definiccedilatildeo 4441 Ao menor valor de 119896 para o qual o grafo 120591(119877) admite uma 119896 minus 119888119900119897119900119903119886ccedilatilde119900

chamamos nuacutemero cromaacutetico de 120591(119877) e indicamos 120594(120591(119877))

Definiccedilatildeo 4442 Dado um grafo 120591(119877) dizemos que 119870 sube 119881(120591(119877)) eacute uma clique de 120591(119877) se

para quaisquer dois veacutertices 119906 119907 isin 119870 tivermos 119906119907 isin 119864(119881) ou seja se o grafo induzido em 120591(119877)

por 119870 eacute um grafo completo Dizemos que 119870 eacute uma clique maximal se todo o 119906 isin 119881(120591(119877))119870 o

conjunto 119870 cup 119906 natildeo eacute uma clique Agrave clique de 120591(119877) com maior cardinalidade chamamos

clique maacutexima e agrave sua cardinalidade nuacutemero de clique de 120591(119877) que indicamos por 120596(120591(119877))

Se 119877 eacute um anel finito entatildeo 120594´ (120591(119877) = ∆(120591(119877))) a menos que 120591(119877) eacute um grafo completo de

ordem iacutempar Observe que o uacutenico grafo completo 120591(119885119899[119894]) ocorre quando 119899 = 1199022 No

entanto neste caso a ordem do grafo eacute 1199022 minus 1 que eacute uniforme entatildeo 120594´(120591(119885119899[119894])) =

∆(120591(119885119899[119894])) Aleacutem disso uma vez que a coloraccedilatildeo de aresta de qualquer grafo leva a uma

coloraccedilatildeo de veacutertice do seu grafo de linha obtemos 120594 (119871(120591(119885119899[119894]))) = ∆(120591(119885119899[119894]))

Claramente 120594(119866) ge 120596(119866) Por outro lado o grafo de linha de 119866 possui um subgrafo completo

de ordem 120549(119866) Assim 120596 (119871(120591(119885119899[119894]))) ge= ∆(120591(119885119899[119894])) Observe que se 119899 =

2119898 119900119906 119902119898 119900119899119889119890 119898 ge 2 entatildeo 120591(119885119899[119894]) tem um veacutertice que eacute adjacente a qualquer outro

veacutertice em 120591(119885119899[119894]) Enquanto se 119899 = 119901119898 119898 ge 1 entatildeo 119885119901119898[119894] cong 119885119901119898 times 119885119901119898 Assim ∆(119885119901119898[119894]) =

1199012119898minus1 minus 1 Isso leva ao seguinte teorema

Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1

Finalmente se 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119904119895 ge 2 119890 119898 119903119895 ge 1 entatildeo o clique e o

nuacutemero cromaacutetico do grafo 119871(120591(119885119899[119894])) eacute dado pelo seguinte Teorema

Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) = (22119898minus1 minus 1) prod (1199011198952119903119895minus1)119903

119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54

Demonstraccedilatildeo O resultado segue calculando ∆(120591(119885119899[119894])) uma vez que ∆(120591(119885119899[119894])) =

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))

Agora encontraremos o diacircmetro do grafo de linha 119871(120591(119885119899[119894])) Primeiro vamos provar que

119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898

Lema 4445 (i) Se 119899 = 2119898 119898 ge 2 entatildeo natildeo haacute um 119886 + 119887119894 119888 + 119889119894 isin 119885119899[119894] onde 119886 119887 119888 119889 satildeo

inteiros iacutempares de modo que (119886 + 119887119894)(119888 + 119889119894) equiv 0(1198981199001198894)

(ii) Se 119899 = 2119898 119898 ge 2 entatildeo natildeo haacute um 119886 + 119887119894 119888 + 119889119894 isin 119885119899[119894] onde 119886 119887 119888 119889 satildeo primos

relativamente com 119902 modo que (119886 + 119887119894)(119888 + 119889119894) equiv 0(119898119900119889119902)

Demonstraccedilatildeo (i) Suponha que (119886 + 119887119894)(119888 + 119889119894) equiv 0(1198981199001198894) Entatildeo 119886119888 minus 119887119889 equiv 0(1198981199001198894) e 119886119889 +

119887119888 equiv 0(1198981199001198894) Visto que 119886 119887 119888 119889 satildeo inteiros iacutempares 119886 = 21198861 + 1

119887 = 21198871 + 1 119888 = 21198881 + 1 119890 119889 = 21198891 + 1para algum 1198861 1198871 1198881 1198891 isin 119885 Assim 119886119888 minus 119887119889 equiv 1198861 + 1198871 +

1198881 + 1198891 equiv 0(1198981199001198892) E 119886119889 + 119887119888 equiv 1198861 + 1198871 + 1198881 + 1198891 equiv 1(1198981199001198892) uma contradiccedilatildeo

(ii) Suponha que (119886 + 119887119894)(119888 + 119889119894) equiv 0(119898119900119889119902) Entatildeo 119886119888 minus 119887119889 equiv 0(119898119900119889119902) e 119886119889 + 119887119888 equiv

0(119898119900119889119902) Visto que 119886 119887 119888 119889 satildeo primos relativamente com 119902 temos 119886 = 1199021198861 + 1198862 119887 = 1199021198871 +

1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim

119886119888 minus 119887119889 equiv 11988621198882 minus 11988721198892 equiv 0(119898119900119889119902) (119868)

119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)

Multiplicando (119868) 119901119900119903 1198882 119890 (119868119868) 119901119900119903 1198892 e adicionando daacute 1198862(11988822 + 1198892

2) equiv 0(119898119900119889119902) Entatildeo

119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e

portanto 1198882 equiv 1198892 equiv 0(119898119900119889119902) uma contradiccedilatildeo ∎

Entatildeo concluiacutemos o seguinte

Teorema 4446 Se 119899 = 2119898 119900119906 119899 = 119902119898 119898 ge 2 119890119899119905atilde119900 119889119894119886119898 (119871(120591(119885119899[119894]))) = 2

Demonstraccedilatildeo (i) Suponha que 119899 = 2119898 119898 ge 2 Entatildeo

1) 119909 = 1198862119905 + 1198872119896119894 onde 119886 119887 119904atilde119900 iacute119898119901119886119903119890119904 119890 119905 ne 119896 119900119906 119905 = 119896 ge lceil1198982rceil implica que 119886119899119899(119909) =

1198882119903 + 1198892119904119894 119888 119890 119889 119904atilde119900 iacute119898119901119886119903119890119904 119890 119903 119904 ge 119898 minus 119898119894119899119905 119896

2) 119909 = 1198862119905(119886 + 119887119894) onde 119886 119887 119904atilde119900 iacute119898119901119886119903119890119904 119890 119905 lt lceil1198982rceil entatildeo 119886119899119899(119909) = 1198882119903 +

1198892119904119894 119888 119890 119889 119904atilde119900 iacute119898119901119886119903119890119904 119890 119903 119904 ge 119898 minus 119905 cup 2119898minus119905minus1(119888 + 119889119894) 119888 119890 119889 119904atilde119900 iacute119898119901119886119903119890119904

55

Aleacutem disso 119889([2119905(1198861 + 1198871119894) 2119898minus119905minus1(1198881 + 1198891119894)] [2119904(1198862 + 1198872119894) 2119898minus119904minus1(1198882 + 1198892119894)]) = 2 se 119905 le 119904 lt

lceil1198982rceil Jaacute que [2119904(1198862 + 1198872119894) 2119898minus119905minus1(1198881 + 1198891119894) ] isin 119881 (119871(120591(119885119899[119894])))

(ii) Suponha que 119899 = 2119898 119898 ge 2 Seja 119909 = 119886119902119905 + 119887119902119896119894 119890 119886 119887 isin 119880(119885119899) Entatildeo 119886119899119899(119909) =

119888119902119903 + 119889119902119904119894 119903 119904 ge 119898 minus 119898119894119899119905 119896 Aleacutem disso 119889([11988611199021199031 + 11988711199021199041119894 11988811199021199051 + 11988911199021198961119894] [11988621199021199032 +

11988721199021199042119894 11988821199021199052 + 11988921199021198962119894] ) = 2 visto que 1199031 1199041 1199052 1198962 ge lceil119898

2rceil o que implica [11988611199021199031 + 11988711199021199041119894 11988811199021199051 +

11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎

Teorema 4447 (i) Se 119899 = 119904119905 119900119899119889119890 119904 119890 119905 119904atilde119900 119889119900119894119904 119901119903119894119898119900119904 119889119894119904119905119894119899119905119900119904 119890 119904 ne 119901 119900119906 119901 ne 119905 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 2

(ii) Se 119899 = 1199041199052 satildeo dois primos distintos e 119904 119905 ne 119901 entatildeo 119889119894119886119898 (119871(120591(119885119899[119894]))) = 2

Demonstraccedilatildeo Primeiro observe que 119871(120591(119877)) ge 2 e para 119899 = 11989911198992 com 119898119889119888(1198991 1198992) = 1

119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]

(i) Caso 1 Se 119899 = 119902119901 119900119906 119899 = 2119901 onde 119901 = 1198862 + 1198872 entatildeo 119881 (119871(120591(119885119899[119894]))) =

[(119906 120572(119886 + 119887119894)) (0 120573(119886 minus 119887119894))] cup [(0 120572(119886 + 119887119894)) (119906 120573(119886 minus 119887119894))] cup [(119906 119907) (0 119907)]

Caso 2 Se 119899 = 2119902 119900119906 119899 = 11990211199021 entatildeo

119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0

(ii) Observe que 119881 (119871(120591(119885119899[119894]))) = [(119906 120572119905) (0 120573119905)] cup [(119906 119907) (0 119907)] 119906 119907 120572 120573 ne 0

∎

Teorema 4448 (i) Se 119899 = 1199041199012 onde 119904 eacute 119901119903119894119898119900 119890 119901 = 1198862 + 1198872 entatildeo 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3

(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3

(iii) Se 119899 = 119901119898119905119897 onde 119901 = 1198862 + 1198872 119898 ge 1 119897 ge 2 e 119898119889119888(119901 119905) = 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3

(iv) Se 119899 = 119904119898119905119897 onde 119904 119905 satildeo primos distintos e 119898 119897 ge 2 entatildeo 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3

Demonstraccedilatildeo (i) Seja 1199071 = [(0 (119886 + 119887119894)2) (1(119886 minus 119887119894)2)] e 1199072 = [(0 (119886 minus 119887119894)(119886 + 119887119894)) (1(119886 minus

119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56

(ii) Seja 119907 = [((1198861 + 1198871119894)119898 (1198862 + 1198872119894)119897 (1198861 minus 1198871119894)119898 (1198862 + 1198872119894)119897)] Entatildeo 119889(119907[(1 0)(0 1)]) = 3

(iii) Seja 119907 = [((119886 + 119887119894)119898 119905) ((119886 minus 119887119894)119898119905119897minus1)] Entatildeo 119889(119907[(1 0)(0 1)]) = 3

(iv) Seja 119907 = [(119904 119905) (119904119898minus1 119905119897minus1 )] Entatildeo 119889(119907[(1 0)(0 1)]) = 3 ∎

Teorema 4449 (i) Se 1198771 1198772 1198773 satildeo corpos e 119877 = 1198771 times 1198772 times 1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 2

(ii) Se 1198771 1198772 1198773 satildeo aneacuteis finitos e 119877119894 natildeo eacute corpo par algum 119894 isin 1 2 3 e 119877 = 1198771 times 1198772 times

1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3

(iii) Se 119877 = prod 119877119894 119900119899119889119890 119896 ge 4 119890119899119905atilde119900 119889119894119886119898 (119871(120591(119877))) = 3119896119894=1

Demonstraccedilatildeo (i) Seja [(1198861 1198862 1198863) (1198871 1198872 1198873) (1198881 1198882 1198883) (1198891 1198892 1198893)] isin 119864 (119871(120591(119877))) Visto que

1198771 1198772 1198773 satildeo corpos (1198861 1198862 1198863) 119900119906 (1198871 1198872 1198873) tem exatamente duas componentes iguais a 0

Seja (1198861 1198862 1198863) = (1198861 0 0) e 1198861 ne 0 Visto que 11988811198891 = 0 1198881 = 0 119900119906 1198891 = 0 Sendo 1198881 = 0 entatildeo

[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2

(ii) Suponha que 1198771 natildeo eacute corpo Seja 119909 119910 isin 1198771lowast de tal modo que 119909119910 = 0 Entatildeo

119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3

(iii) Seja 119909 = (119909119895) 119900119899119889119890 119909119895 = 1 119904119890 119895 = 1 2 119890 0 de outra forma 119910 = (119910119895) onde 119910119895 = 1 se 119895 =

3 4 119890 0 de outra forma 119911 = (119911119895) onde 119911119895 = 1 se 119895 = 2 3 119890 0 de outra forma e 119908 = (119908119895) onde

119908119895 = 1 se 119895 = 1 4 119890 0 de outra forma Entatildeo 119889([119909 119910] [119911 119908]) = 3 ∎

Teorema 44410 (i) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 se e somente se 119899 =

119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886

445 Cintura e Raio de L(τ(Zn[i]))

Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3

Lema 4452 Se existe um veacutertice 119907 isin 119871(120591(119885119899[119894])) com excentricidade 2 entatildeo

119903119886119889 (119871(120591(119885119899[119894]))) = 2

Demonstraccedilatildeo Observe que 119871(120591(119885119899[119894])) natildeo tem grafo de estrelas abrangente jaacute que se

119886 119887 isin 119881(120591(119885119899[119894])) de tal modo que 119886 ne 119887 119890 119886119887 = 0 119890119899119905atilde119900 119889([119886 119887] [119886119894 119887119894]) gt 1 ∎

Teorema 4453 Se 119899 = 2119898 119899 = 119902119898 119898 ge 2 119900119906 119899 = 119901119898 119898 ge 1 entatildeo 119903119886119889 (119871(120591(119885119899[119894]))) = 2

57

Demonstraccedilatildeo (1) Se 119899 = 2119898 119898 ge 2 entatildeo 119889([2119898minus1 + 2119898minus1119894 2][119909 119910]) le 2 para todo [119909 119910] isin

119881 (119871(120591(119885119899[119894])))

(2) Se 119899 = 119902119898 119898 ge 2 entatildeo 119889([119902119898minus1 119902] [119909 119910]) le 2 para todo [119909 119910] isin 119881 (119871(120591(119885119899[119894])))

(3) Se 119899 = 119901119898 119898 ge 1 entatildeo 119889([(119886 + 119887119894)119898(119886 minus 119887119894)119898minus1 (119886 minus 119887119894)119898(119886 + 119887119894)119898minus1] [119909 119910]) le 2 para

todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎

Teorema 4454 Se 119899 = 119903119898119905 onde 119903 = 2 119902 119900119906 119901 119890 119898 ge 1 119898119889119888(119903 119905) = 1 entatildeo

119903119886119889 (119871(120591(119885119899[119894]))) = 2

Demonstraccedilatildeo (1) Se 119903 = 2 119900119906 119902 entatildeo 119889([(119903119898minus1 0) (119903 1)] [(119909 119910) (119905 119904)]) le 2 para todo

[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))

(2) Se 119903 = 119901 = 1198862 + 1198872 entatildeo 119889 = ([((119886 + 119887119894)119898(119886 minus 119887119894)119898minus1 0) ((119886 minus 119887119894)119898(119886 +

119887119894)119898minus1 0) ] [(119909 119910) (119905 119904)]) le 2 para todo [(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894]))) ∎

Resumindo os resultados acima obtemos o seguinte

Teorema 4185 O raio do grafo de linha 119871(120591(119885119899[119894])) eacute igual a 2

446 O Nuacutemero de Dominaccedilatildeo de L(τ(Zn[i]))

Nesta seccedilatildeo determinamos o nuacutemero de dominaccedilatildeo de 119871(120591(119885119899[119894])) quando 119899 = 119905119898 e 119905 eacute

primo

O estudo do nuacutemero de dominaccedilatildeo do grafo de linha de 119866 leva ao estudo do nuacutemero de

dominacircncia de linha ou linha de 119866 isto eacute 120574(119871(119866)) = 120574 ´(119866) Por outro lado para qualquer

grafo 119866 120574119894´(119866) = 120574 ´(119866) Aleacutem disso se 119866 eacute o grafo bipartido completo 119870119903119904 entatildeo 120574 ´(119866) =

min (119903 119904) entatildeo temos o seguinte

Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022

Agora estudamos o nuacutemero de dominaccedilatildeo do grafo de linha de 120591(119885119899[119894]) quando 119899 eacute uma

potecircncia de um primo O primeiro teorema trata o caso 119899 = 2119898 119898 ge 2 Aqui fazemos uso do

fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1

2(2119898 minus 1)rfloor

Demonstraccedilatildeo Para 119895 = 1 2 hellip 2119898 minus 1 seja 1198601 = 12057222119898minus119895 120572 isin 1 3 hellip 2119895 minus 1 Observe que

os conjuntos 119860119895 formam uma particcedilatildeo para os veacutertices de 120591(11988522119898) Seja 119878 = ⋃ 119860119895119898119895=1 119890 119879 =

⋃ 1198601198952119898minus1119895=119898+1 Entatildeo o conjunto 119878 induz um subgrafo completo de 120591(11988522119898) e o conjunto 119879 forma

um conjunto independente dele E cada veacutertice em 119860119896 eacute adjacente a cada veacutertice em

⋃ 1198601198952119898minus119896119895=1 120591(11988522119898) natildeo tem outras arestas Seja 119863 sub 119864(120591(11988522119898)) ser um conjunto dominante de

veacutertices para 119871(120591(11988522119898)) com cardinalidade miacutenima Como o conjunto 119878 induz um subgrafo

completo de 120591(11988522119898) da ordem 2119898 minus 1 entatildeo 120574 (119871(120591(1198852119898[119894]))) ge lfloor1

2(2119898 minus 1)rfloor Por outro lado

uma vez que 119863 domina todas as arestas no grafo completo lang119878rang 119863 tambeacutem domina todas as

arestas juntando 119878 119901119886119903119886 119879 lembre-se de que 119879 forma um conjunto independente e portanto

120574 (119871(120591(1198852119898[119894]))) = lfloor1

2(2119898 minus 1)rfloor ∎

A prova do Teorema 4462 mostra o conjunto 119879 eacute um conjunto independente com

cardinalidade maacutexima em 120591(1198852119898[119894]) enquanto o conjunto 119878 induz um subgrafo completo com

a ordem maacutexima

Entatildeo o seguinte corolaacuterio eacute obtido

Corolaacuterio 4463 Para 119899 = 2119898 119898 ge 2

(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)

Como outra consequecircncia para a prova do teorema anterior obteacutem-se o seguinte corolaacuterio

que daacute a sequecircncia de grau para 120591(1198852119898[119894])

Corolaacuterio 4464 Para 119895 = 1 2 hellip 2119898 minus 1 o grafo 120591(1198852119898[119894]) tem exatamente 2119895minus1 veacutertices

de grau 22119898minus119895 minus 2 se 1 le 119895 le 119898 e 2119895minus1 veacutertices de grau 22119898minus119895 minus 1 se 119898 + 1 le 119895 le 2119898 minus 1

Demonstraccedilatildeo Para cada 119907 isin 119860119895 onde 1 le 119895 le 119898 1199072 = 0 assim 119889119890119892(119907) = |⋃ 1198601198962119898minus119895119896=1 | minus 1 =

22119898minus119895 minus 1 E para cada 119907 isin 119860119896 onde 119898 + 1 le 119896 le 2119898 minus 1 1199072 ne 0 assim 119889119890119892(119907) = |⋃ 1198601198962119898minus119895119896=1 | =

22119898minus119895 minus 1 ∎

Aleacutem disso a prova do teorema acima mostra que a excentricidade de 22119898minus1 eacute 1 e a

excentricidade de qualquer outro veacutertice em 120591(11988522119898) eacute 2 uma vez que o veacutertice 2 eacute

adjacente apenas ao veacutertice 22119898minus1 e para qualquer 119909 isin 119881(120591(1198852119898[119894])) 2 minus 22119898minus1 minus 119909 eacute um

caminho do comprimento 2 Isso leva ao seguinte corolaacuterio

Corolaacuterio 4465 O centro do grafo 120591(1198852119898[119894]) eacute o conjunto 2119898minus1(1 + 119894)

59

Em seguida encontramos o nuacutemero de dominaccedilatildeo do grafo de linha 119871(120591(119885119899[119894])) onde 119899 =

119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2

1) Se 119860119896119895 = 119886119902119896 + 119887119902119895119894 119886 isin 119880(119885119902119898minus119896) 119887 isin 119880(119885119902119898minus119896) 119890119899119905atilde119900 |119860119896119895| = (119902 minus 1)21199022119898minus119896minus119895minus2

quando 1 le 119896 119895 le 119898 minus 1 |119860119898119895| = 119902119898minus119895 minus 119902119898minus119895minus1 e |119860119896119898| = 119902119898minus119896 minus 119902119898minus119896minus1 onde 119896 119895 ne

119898

2) Se 119878 = (⋃ 119860119896119895lceil1198982rceille119896119895le119898 ) minus 119860119898119899 119890119899119905atilde119900 |119878| = 1199022lceil1198982rceil minus 1

(ii) Para 119898 ge 3 se 119879 = ⋃ 1198601198961198951le119896119895lelceil119898

2rceilminus1

119890119899119905atilde119900 |119879| = 1199022lceil1198982rceil(119902lceil1198982rceil minus 1)2

Teorema 4467 Se 119899 = 119902119898 119898 ge 2 entatildeo 120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) =

(1

2) (119902119898 minus 1) se 119898 eacute par e (

1

2) (1199022lfloor1198982rfloor + 1) 119904119890 119898 eacute iacute119898119901119886119903

Demonstraccedilatildeo Seja 119860119896119895 119878 119890 119879 definidos como dados no Lema 4466 Claramente o

conjunto 119878 induz um subgrafo completo de 120591(119885119899[119894]) com ordem maacutexima se 119898 for par e 119878 cup

119902lfloor1198982rfloor induz um subgrafo completo de 120591(119885119899[119894]) com ordem maacutexima se 119898 for iacutempar Por

outro lado se 119898 ge 3 entatildeo 119879 formam um conjunto independente com cardinalidade maacutexima

Aleacutem disso se um veacutertice 119907 pertence ao conjunto 119860119903119904 entatildeo 119907 eacute adjacente a cada elemento

em 119860119896119895 onde 119898 minus 119898119894119899119903 119904 le 119896 119895 le 119898 e 119896 119895 ne 119898 ao mesmo tempo 120591(119885119899[119894]) natildeo tem outras

arestas ∎

Como consequecircncia da prova do Teorema 4462 concluiacutemos o seguinte

Corolaacuterio 4468 Se 119899 = 119902119898 119898 ge 2 entatildeo

(i) 120596(120591(119885119899[119894])) = 119902119898 minus 1 119904119890 119898 eacute 119901119886119903 119890 1199022lfloor1198982rfloor 119904119890 119898 eacute iacute119898119901119886119903

(ii) 120573 (119871(120591(119885119899[119894]))) = 1 119904119890 119898 = 2 119890 120573 (119871(120591(119885119899[119894]))) = 1199022lfloor1198982rfloor(119902lceil1198982rceil minus 1)2 se 119898 ge 3

Corolaacuterio 4469 Seja 119899 = 119902119898 119898 ge 2 e 119907 = 119886119902119903 + 119887119902119904119894 onde 119886 119887 isin 119880(119885119899) Entatildeo

119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil

Corolaacuterio 44610 Seja 119899 = 119902119898 119898 ge 2 Entatildeo

60

(i) A excentricidade de cada 119907 isin 119860(119898minus1)(119898minus1) eacute 1 e a excentricidade de qualquer outro

veacutertice 119907 isin 120591(119885119899[119894]) eacute 2

(ii) O centro do grafo 120591(119885119899[119894]) eacute o conjunto 119860(119898minus1)(119898minus1)

(iii) O raio do grafo 120591(119885119899[119894]) eacute igual a 1

(iv) O diacircmetro do grafo 120591(119885119899[119894]) eacute igual a 2 para 119898 ge 3

Finalmente encontramos o nuacutemero de dominaccedilatildeo do grafo de linha 119871(120591(119885119899[119894])) onde 119899 =

119901119898 119898 ge 2

Note que 119885119901119898[119894] cong 119885119901119898 times 119885119901119898 Seja 119860119896119895 = (119886119901119896 119887119901119895) 119886 isin 119880(119885119901119898minus119896) 119887 isin 119880(119885119901119898minus119895) Claramente

o conjunto 119860119896119895 0 le 119896 119895 le 119898 e natildeo ambos 119896 119895 = 119898 119900119906 0 particcedilatildeo de veacutertices de 120591(119885119901119898 times 119885119901119898)

Lema 44611 (i) Para 119898 ge 2

1) Se 119878 = (⋃ 119860119896119895lceil1198982rceille119896119895le119898 ) minus 119860119898119898 119890119899119905atilde119900 119904 = |119878| = 1199012lfloor1198982rfloor minus 1

2) Se 1198711 = ⋃ 1198601198961198980le119896lelceil1198982rceilminus1 e 1198712 = ⋃ 1198601198961198980le119896lelceil1198982rceilminus1 entatildeo 119897 = |1198711| = |1198712| = 119901119898 minus 119901lfloor1198982rfloor

(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896

lceil1198982rceilminus1119896=1 entatildeo 119887 = |119861| = (119901119898 minus 119901119898minus1)(lceil1198982rceil minus 1) minus (119901119898minus1 minus

119901lfloor1198982rfloor)

2) Se 119879 = ⋃ 119860119896119895 minus 119860000le119896lelceil1198982rceilminus1 entatildeo 119905 = |119879| = (119901119898minus1 minus 119901lfloor1198982rfloor)2

+ 2(119901 minus 1)(1199012119898minus2 minus

1199012119898minuslceil1198982rceilminus1)

(iii) Para 119898 ge 4

Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1

119898minus1119896=lceil1198982rceil 1198822 = ⋃ ⋃ 119860119896119895

119898minus119896minus1119896=1

119898minus1119895=lceil1198982rceil e 119882 = 1198821 cup 1198822 entatildeo 119908 = |119882| =

2119901119898minus1 ((119901lfloor1198982rfloor minus 1) minus lfloor1198982rfloor(119901 minus 1))

Teorema 44612 Seja 119899 = 119901119898 119898 ge 2 e 119904 119897 119890 119887 definidos no Lema 472 entatildeo

120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar

Demonstraccedilatildeo Usando algumas noccedilotildees do Lema 44611 Observe que o conjunto 119878 induz

um subgrafo completo de 120591(119885119899[119894]) 119870119904 Assim qualquer conjunto de dominaccedilatildeo de aresta para

120591(119885119901119898 times 119885119901119898) deve conter 1199042 arestas para dominar 119870119904 Se 119898 ge 3 o conjunto 119871 = 1198711 cup 1198712

induz um grafo bipartido completo 119870119897119897 com conjuntos bipartidos 1198711 119890 1198712 Isso contribui com as

arestas na margem dominante definidas para 120591(119885119901119898 times 119885119901119898)

61

As arestas que juntam veacutertices em 119870119897119897 aos veacutertices em 119870119904 satildeo cobertos pelos mesmos

conjuntos dominantes de aresta para 119870119897119897 e 119870119904 Aleacutem disso os veacutertices em 1198601198960 e 1198600119896 onde 1 le

119896 le 119898 minus 1 satildeo apenas adjacentes a alguns veacutertices em 119870119904 e 119870119897119897

Por outro lado se 119898 ge 3 o conjunto 119879 eacute um conjunto independente Felizmente os veacutertices

em 119879 satildeo apenas adjacentes a veacutertices em 119878 Assim qualquer conjunto de dominacircncia de

aresta para 119870119904 tambeacutem domina arestas entre 119878 119890 119879

Agora para cada 1 le 119896 le lceil1198982rceil minus 1 119890 119898 minus 119896 le 119895 le 119898 o conjunto 119860119896119895 cup 119860119895119896 induz um grafo

bipartido completo com conjuntos bipartidos 119860119896119895 119890 119860119895119896 Para dominar esta coleccedilatildeo de grafos

bipartidos completos induzidos por 119860119896119895 cup 119860119895119896 precisamos de arestas 119887 aresta no conjunto

dominante de arestas para 120591(119885119901119898 times 119885119901119898) Felizmente esse conjunto dominante com

elementos 119887 tambeacutem domina todas as arestas em 119864 (120591(119885119901119898 times 119885119901119898)) que satildeo incidentes de

qualquer aresta nesta coleccedilatildeo

Finalmente observe que se 119898 ge 4 os veacutertices em 119882 satildeo apenas adjacentes a alguns veacutertices

em 119870119904 bem como na coleccedilatildeo dos grafos bipartidos completos O grafo 120591(119885119901119898 times 119885119901119898) natildeo tem

outras arestas ∎

62

CAPIacuteTULO 5

GRAFO EQUILIBRADO DOS DIVISORES DE ZERO DE ANEacuteIS DE MATRIZ

Neste uacuteltimo capiacutetulo tratamos de aneacuteis natildeo comutativos Neste caso em vez de um grafo

simples passamos a ter um grafo orientado

Definiccedilatildeo 51 Grafo Orientado consiste em um grafo 119866 = (119881 119860) onde 119881 = 1199071 hellip 119907119899 eacute um

conjunto de veacutertices e 119860 = 1198861 hellip 119886119896 eacute um conjunto de arcos tais que 119886119896 119896 = 1 hellip 119898 eacute

representado por um par ordenado (119907119894 119907119895) de veacutertices 119894 119895 = 1 hellip 119899

Definiccedilatildeo 52 O conjunto de arcos que saem de um veacutertice 119907 eacute chamado de leque

Definiccedilatildeo 53 O grau de saiacuteda de um veacutertice 119907 eacute nuacutemero de arcos que saem de 119907 ou seja o

tamanho do leque de saiacuteda de 119907

Definiccedilatildeo 54 O grau de entrada de um veacutertice 119907 eacute o nuacutemero de arcos que terminam em 119907

Definiccedilatildeo 55 Um grafo diz-se equilibrado se para todos os veacutertices o grau de saiacuteda eacute igual ao

grau de entrada

Vamos revisar brevemente algumas definiccedilotildees e ferramentas que seratildeo usadas

posteriormente Seja 119877 um anel natildeo comutativo O grafo de divisores de zero dirigido de 119877 eacute

um grafo dirigido 120591(119877) com o conjunto de veacutertices 119885(119877)lowast = 119885(119877) minus 0 onde para distintos

veacutertices 119909 e 119910 de 119885(119877)lowast existe uma aresta dirigida 119909 rarr 119910 de 119909 para 119910 se e somente se 119909119910 = 0

Se 119883 eacute um subconjunto de um anel 119877 entatildeo o anulador esquerdo de 119883 eacute 119886119899119899119897(119883) =

119886 isin 119877 119886119883 = 0 e o anulador direito de 119883 eacute 119886119899119899119903(119883) = 119886 isin 119877 119883119886 = 0 Se o anel 119877 eacute

comutativo entatildeo obviamente 119886119899119899119897(119883) = 119886119899119899119903(119883) Neste caso escrevemos simplesmente

119886119899119899(119883) Observe que se 119877 eacute um anel finito e 119886 isin 120591(119877) entatildeo o grau de entrada de um veacutertice eacute

|119886119899119899119897(119886)| minus 1 119904119890 1198862 ne 0 119890 |119886119899119899119897(119886)| minus 2 119904119890 1198862 = 0 e de forma semelhante para o grau de saiacuteda

Aleacutem disso o grau de 119886 no grafo natildeo dirigido 120591(119877) eacute |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 1 119904119890 1198862 ne

0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0

Em [3] Akabari e Mohammadian provaram que para cada corpo finito 119865 119890 119899 ge 2 o grau de

entrada e o grau de saiacuteda de um veacutertice 119886 isin 120591(119872119899(119865)) satildeo |119865|119899(119899minus119896) minus 휀 e o grau de 119886 isin

120591(119872119899(119865)) eacute 2|119865|119899(119899minus119896) minus |119865|(119899minus119896)2minus 휀 119900119899119889119890 휀 = 1 a menos que 1198862 = 0 e neste caso 휀 = 2 Em

particular o grafo dirigido 120591(119872119899(119865)) eacute equilibrado Vamos ver que este resultado pode ser

estendido ao anel das matrizes sobre um anel comutativo com identidade e de ideais

principais

63

51 Resultados auxiliares

Para provar os principais resultados utilizamos a teoria de anel de divisatildeo elementar Esta

teoria diz respeito a reduccedilatildeo de matrizes para uma forma diagonal Uma matriz 119899 por 119898 119860 =

(119886119894119895) eacute dita ser diagonal se 119886119894119895 = 0 119901119886119903119886 119905119900119889119900119904 119894 ne 119895 Dizemos que uma matriz 119860 sobre um anel

119877 admite reduccedilatildeo diagonal se existir as matrizes invertiacuteveis 119875 e 119876 119904119900119887119903119890 119877 de modo que 119875119860119876

eacute uma matriz diagonal Duas matrizes 119860 119890 119861 sobre um anel 119877 dizem-se equivalentes

(notaccedilatildeo 119860 ~ 119861) se houver matrizes invertiacuteveis 119875 119890 119876 tal que 119861 = 119875119860119876 Seguindo Kaplansky

[28] se toda matriz 119860 sobre um anel 119877 eacute equivalente a uma matriz diagonal 119863119860 =

119889119894119886119892 (1198891 119889119899) com a propriedade que 119889119894 eacute um divisor total de 119889119894+1 entatildeo 119877 eacute chamado de

anel de divisatildeo elementar Os elementos 1198891 119889119899 satildeo chamados divisores elementares da

matriz 119860 Os aneacuteis de divisatildeo elementar foram estudados por muitos autores [24 39 40 41]

O seguinte teorema fundamental que eacute provado em [18] fazendo uso de dois teoremas de

Kaplansky

Teorema 511 Qualquer anel comutativo de ideais principais com identidade eacute um anel de

divisatildeo elementar

Lembre-se de que um anel comutativo de ideal principal eacute um anel comutativo em que todo

ideal eacute um ideal principal

No Lema seguinte mostramos que em aneacuteis finitos o tamanho dos anuladores eacute invariante sob

multiplicaccedilatildeo por unidades

Lema 512 Seja 119877 um anel finito com identidade Se 119906 isin 119877 eacute uma unidade entatildeo para

qualquer 119886 isin 119877 temos

|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|

Demonstraccedilatildeo Obviamente 119886119899119899119897(119886) sube 119886119899119899119897(119886119906) Suponha que 119908 isin 119886119899119899119897(119886119906) Como 119906 eacute uma

unidade a equaccedilatildeo 119908119886119906 = 0 implica que 119908119886 = 0 Portanto 119908 isin 119886119899119899119897(119886) e depois 119886119899119899119897(119886119906) =

119886119899119899119897(119886)

Eacute faacutecil mostrar que se 120593 119877 rarr 119877 eacute um automorfismo e 119909 eacute um elemento de 119877 temos que

120593(119886119899119899119897(119909)) = 119886119899119899119897(120593(119909)) o que implica que |119886119899119899119897(119909)| = |119886119899119899119897(120593(119909))| Como o mapa 120593 119877 rarr 119877

definido por 120593(119903) = 119906minus1119903119906 eacute um automorfismo e 120593(119906119886) = 119886119906 segue que |119886119899119899119897(119906119886)| =

|119886119899119899119897(119886119906)| Isso prova o lema ∎

Observaccedilatildeo 1 Eacute faacutecil verificar se um resultado semelhante eacute vaacutelido para anuladores direitos

52 Resultados principais

64

O seguinte teorema eacute uma generalizaccedilatildeo do Lema 14 provado por Akabari e Mohammadian em

[3]

Teorema 521 Seja 119877 um anel comutativo de ideal principal finito com identidade e 119899 ge 2

Suponha que 119860 isin 119872119899(119877) eacute um divisor de zero diferente de zero e 1198891 1198892 119889119899 sejam divisores

elementares de 119860 Entatildeo o grau de entrada e o grau de saiacuteda de 119860 119890119898 120591(119872119899(119877)) satildeo

2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1

e o grau de 119860 119890119898 120591(119872119899(119877)) eacute igual a

2 prod|119886119899119899(119889119894)|119899 minus prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)| minus 휀

119899

119894119895=1

119899

119894=1

onde 휀 = 1 a menos que 1198602 = 0 e neste caso 휀 = 2 Em particular 120591(119872119899(119877)) eacute equilibrado

Demonstraccedilatildeo Pelo Teorema 511 o anel 119877 eacute um anel de divisatildeo elementar Entatildeo

suponha que 119860 ~ 119889119894119886119892(1198891 119889119899) = 119863119860 Resulta do Lema 512 e da Observaccedilatildeo 1 que

|119886119899119899119897(119860)| = |119886119899119899119897(119863119860)| 119890 |119886119899119899119903(119860)| = |119886119899119899119903(119863119860)| Agora para uma matriz 119883 = (119909119894119895) no anel de

matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)

Como o anel 119877 eacute comutativo segue-se que

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

Por isso o grau de entrada e o grau de saiacuteda de 119860 satildeo ambos iguais a prod |119886119899119899119897(119889119894)|119899119894=1

119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2

Das equaccedilotildees (1) e (2) concluiacutemos que uma matriz 119883 = (119909119894119895) pertence a 119886119899119899119897(119863119860) cap 119886119899119899119903(119863119860)

se e somente se 119909119894119895 isin 119886119899119899(119889119894) cap 119886119899119899(119889119895) Consequentemente

|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119886119899119899(119889119894)|119899 minus prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

119899

119894=1

Assim o grau de 119860 119890119898 120591(119872119899(119877)) eacute 2 prod |119886119899119899(119889119894)|119899 minus prod |119886119899119899(119889119894) cap 119886119899119899(119889119895)| minus 휀119899119894119895=1

119899119894=1 onde

119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2 Isso completa a prova∎

Observaccedilatildeo 2 Suponha que o anel 119877 seja um corpo finito 119865 Entatildeo na reduccedilatildeo diagonal 119863119860 =

119889119894119886119892(1198891 119889119899) da matriz 119860 cada divisor elementar eacute zero ou uma unidade Aleacutem disso o

nuacutemero de divisores elementares diferentes de zero eacute igual ao rank de 119860 Note que 119886119899119899(119889119894) =

0 se 119889119894 eacute uma unidade e 119886119899119899(119889119894) = 119865 se 119889119894 for zero Assim denotando por 119896 o rank de 119860

temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2

Entatildeo obtemos os resultados de Akabari e Mohammadian provado em [1]

Um circuito num grafo dirigido eacute uma sequecircncia 1198901 1198902 119890119903 de arestas dirigidas distintas de

modo que o veacutertice final de 119890119894 eacute o veacutertice inicial de 119890119894+1 para todos 1 le 119894 le 119903 minus 1 e o veacutertice

final de 119890119903 eacute o veacutertice inicial de 1198901 Um circuito eacute Euleriano se incluir cada aresta exatamente

uma vez e visitar todos os veacutertices Um grafo dirigido eacute chamado Euleriano se conteacutem um

circuito Euleriano

Teorema 522 Seja 119877 um anel comutativo de ideal principal finito com identidade e 119899 ge

2 Entatildeo o grafo do divisor de zero dirigido 120591(119872119899(119877)) eacute Euleriano

Demonstraccedilatildeo Seja 119885119897(119872119899(119877)) o conjunto de divisores de zero esquerdo de 119872119899(119877) Ou seja

119885119897(119872119899(119877)) = 119883 isin 119872119899(119877) 119883119860 = 0 para alguns 119860 isin 119872119899(119877) minus 0 Da mesma forma seja

119885119903(119872119899(119877)) o conjunto dos divisores de zero direito de 119872119899(119877) Em [31] foi mostrado que

119885119897(119872119899(119877)) = 119885119903(119872119899(119877)) Por outro lado Redmond mostrou em [44] que para um anel natildeo

comutativo o grafo do divisor de zero dirigido eacute conexo se e somente se o conjunto de

divisores de zero direito eacute igual ao conjunto de divisores de zero esquerdo Portanto

120591(119872119899(119877)) eacute conexo Para completar a prova observamos que um grafo dirigido conexo eacute

Euleriano se for equilibrado em [23]∎

Em seguida determinamos o menor anel comutativo finito com identidade tal que 120591(119872119899(119877))

natildeo eacute equilibrado

66

Teorema 523 Seja 119877 um anel comutativo com identidade e 119899 ge 2 Se o grafo do divisor de

zero dirigido 120591(119872119899(119877)) natildeo eacute equilibrado entatildeo |119877| ge 8 Aleacutem disso existe um anel

comutativo com identidade da ordem 8 tal que 120591(119872119899(119877)) natildeo eacute equilibrado

Demonstraccedilatildeo Para qualquer inteiro positivo 119899 seja 120574(119899) o nuacutemero de aneacuteis a menos de

isomorfismo da ordem 119899 (incluindo aneacuteis natildeo-comutativos e aneacuteis sem identidade) Usando a

classificaccedilatildeo de grupos abeliano finito e o fato de que quando se decompotildee o grupo aditivo de

um anel finito em suas componentes primaacuterias os componentes satildeo ideais de ordem de

potecircncia de um primo concluiacutemos que 120574 eacute multiplicativa Se 119899 = 11990111198901 119901119896

119890119896 eacute a factorizaccedilatildeo

prima de 119899 entatildeo 120574(119899) = 120574( 11990111198901) 120574(119901119896

119890119896)

O anel nulo eacute o uacutenico anel com um elemento e natildeo tem identidade Suponha que 119877 eacute um anel

finito da ordem 119901 onde 119901 eacute 119906119898 119899uacute119898119890119903119900 119901119903119894119898119900 Lembre-se de que cada elemento em um anel

finito eacute uma unidade ou um divisor de zero Portanto se 119877 natildeo possui divisores de zero

diferentes de zero entatildeo cada elemento diferente de zero eacute uma unidade Por isso 119877 eacute um

anel de divisatildeo finita e assim pelo pequeno teorema de Wedderburn 119877 eacute isomorfo para o

corpo primo 119865119901 Em seguida suponha que 119877 tem divisores de zero diferentes de zero Seja 119903 isin

119877 um divisor de zero diferente de zero Uma vez que o grupo aditivo de 119877 tem a ordem 119901 pelo

teorema de Lagrange natildeo possui subgrupos natildeo triviais Como 119886119899119899119897(119903) eacute um subgrupo do grupo

aditivo de 119877 devemos ter 119886119899119899119897(119903) = 119877 Portanto todos os elementos de 119877 satildeo divisores de

zero Assim para cada elemento 119903 isin 119877 temos 119886119899119899119897(119903) = 119886119899119899119903(119903) = 119877 Isso implica claramente

que 119877 eacute o anel nulo da ordem 119901 Entatildeo 120574(119901) = 2 Observe que o anel zero natildeo possui

identidade e 120591(119872119899(119865119901)) eacute equilibrado pelo Teorema 521

Agora sejam 119901 119890 119902 nuacutemeros primos distintos Como 120574 eacute multiplicativo 119905119890119898119900119904 120574(119901119902) = 4 Se 119877

for um anel da ordem 119901119902 entatildeo 119877 tem um 1198681 ideal de ordem 119901 e um ideal 1198682 de ordem 119902

Como 119901 119890 119902 satildeo primos distintos temos 119877 cong 1198682 oplus 1198682 Portanto se 119877 tem identidade devemos

ter 119877 cong 119865119901 oplus 119865119902 onde 119865119901 119890 119865119902 satildeo corpos primos de ordem 119901 119890 119902 respectivamente Como 119865119901 oplus

119865119902 eacute um anel de ideal principal o grafo 120591(119872119899(119865119901 oplus 119865119902)) eacute equilibrado pelo Teorema 521

Agora suponha que 119877 tenha ordem 1199012 onde 119901 eacute um nuacutemero primo Existem 11 aneacuteis de ordem

1199012 [14] No entanto se um anel 119877 de ordem 1199012 eacute comutativo com identidade eacute

necessariamente um anel de ideal principal De fato todo ideal proacuteprio 119868 diferente de zero

em particular tem ordem 119901 Por isso 119868 eacute gerado como um grupo aditivo por qualquer

elemento diferente de zero

Uma vez que para 1 lt 119899 lt 8 temos que 119899 eacute primo um quadrado de primo ou o produto de

dois primos concluiacutemos que 120591(119872119899(119877)) eacute equilibrado para |119877| lt 8

67

Para completar a prova deixe-nos dar um exemplo de um anel 119877 de ordem 8 de modo que

119872119899(119877) natildeo seja equilibrado Seja 119877 a aacutelgebra tridimensional sobre o corpo 1198652 com base

1 119886 119887 e a seguinte multiplicaccedilatildeo da tabela

1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0

Observe que o anel 119877 natildeo eacute um anel de ideal principal Por exemplo o ideal (a b) natildeo eacute

principal Uma simples computaccedilatildeo mostra que para o veacutertice 119860 = [119886 0119887 0

] no grafo 120591(1198722(119877)) o

grau de entrada eacute 254 e o grau de saiacuteda eacute 1022 Portanto o grafo 120591(1198722(119877)) natildeo eacute equilibrado

∎

68

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ii

iii

Dedicatoacuteria




iv

v

Agradecimentos







me















vi

vii

Resumo







Palavras Chave


grafo de linha

viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






iii

Dedicatoacuteria




iv

v

Agradecimentos







me















vi

vii

Resumo







Palavras Chave


grafo de linha

viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






iv

v

Agradecimentos







me















vi

vii

Resumo







Palavras Chave


grafo de linha

viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






v

Agradecimentos







me















vi

vii

Resumo







Palavras Chave


grafo de linha

viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






vi

vii

Resumo







Palavras Chave


grafo de linha

viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

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(1949)

70



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(1995)














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71






vii

Resumo







Palavras Chave


grafo de linha

viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






viii

ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






ix

Abstract







Keywords


graph

x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






x

xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






xi










119903(119866) ou 119903119886119889(119866)- Raio do grafo G






119886119899119899(119909)- Anulador de x








xii








xiii

xiv


CAPIacuteTULO 1 3




CAPIacuteTULO 2 15




CAPIacuteTULO 3 23



32 Exemplos 24

33 Propriedades de 120533(119825) 25

34 Automorfismo de 120533(119825) 32

CAPIacuteTULO 4 35
















CAPIacuteTULO 5 62




xv

BIBLIOGRAFIA 68

1










comutativos em [30]






















2

3

CAPIacuteTULO 1

























uacuteltimo assunto







4

















Teorema de Fermat



119909119899 + 119910 119899 = 119911119899


assunto











5

119909 + 119910 = 119910 + 119909


(119909 + 119910) + 119911 = 119909 + (119910 + 119911)






(119909 ∙ 119910) ∙ 119911 = 119909 ∙ (119910 ∙ 119911)


esquerda a saber

119909 ∙ (119910 + 119911) = 119909 ∙ 119910 + 119909 ∙ 119911 119890 (119910 + 119911) ∙ 119909 = 119910 ∙ 119909 + 119911 ∙ 119909











ou unidades de A




respectivamente

6



I 119909 minus 119910 isin 119861

II 119909 ∙ 119910 isin 119861


119861 = 0 3 6


reais

Note que









119860 e para cada 119909 isin 119868 119905119890119898119900119904

119886 ∙ 119909 isin 119868 119890 119909 ∙ 119886 isin 119868


do anel


De facto


7

119909 = 0 rarr 0 ∙ 0 = 0 isin 119868

0 ∙ 1 = 0 isin 119868

0 ∙ 2 = 0 isin 119868

0 ∙ 3 = 0 isin 119868

119909 = 2 rarr 2 ∙ 0 = 0 isin 119868

2 ∙ 1 = 2 isin 119868

2 ∙ 2 = 0 isin 119868







119909 isin 2119885 119909 = 2119898 119898 isin 119885


O conjunto 1198722 = (119886 119887119888 119889


unidade

O conjunto 119861 = (119886 1198870 0


ideal de A




Propriedades


8



unidade u








Gauss
















119911 ∙ 119908 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894

119908 ∙ 119911 = (119888119886 minus 119889119887) + (119889119886 + 119888119887)119894 = (119886119888 minus 119887119889) + (119886119889 + 119887119888)119894 = 119911 ∙ 119908



119911 ∙ 119908 = 0 rarr |119911 ∙ 119908| = 0

|119911| ∙ |119908| = 0 rarr |119911| = 0 119900119906 |119908| = 0

9

|119911| = 0 harr 119911 = 0 119890

|119911| = radic1198862 + 1198872 ∎








1199092 + 1199102 = (119909 minus 119910119894)(119909 + 119910119894)

119937[119946] e sua norma




119873(119911) = 119911119911 = (119886 + 119887119894)(119886 minus 119887119894) = 119886 2 + 1198872


moacutedulo

|119886 + 119887119894| = radic1198862 + 1198872 119873(119886 + 119887119894) = 1198862 + 1198872 = |119886 + 119887119894|2







Entatildeo temos

119873(119911)119873(119908) = (1198862 + 1198872)(1198882 + 1198892) = (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (1)

119873(119911119908) = (119886119888 minus 119887119889)2 + (119886119889 + 119887119888)2

= (119886119888)2 minus 2119886119888119887119889 + (119887119889)2 + (119886119889)2 + 2119886119889119887119888 + (119887119888)2

10

= (119886119888)2 + (119886119889)2 + (119887119888)2 + (119887119889)2 (2)


119873(119911119908) = 119873(119911)119873(119908) ∎



119873(119886 + 119887119894) = |119886 + 119887119894|2 = 1198862 + 1198872




120574 = 120572120573 119901119886119903119886 119886119897119892119906119898 120573 isin 119885 [119894]

Entatildeo

119873(120574) = 119873(120572)119873(120573)

isto eacute a 119873(120572) 119889119894119907119894119889119890 119886 119873(120574)





em 119885 Em [29]









Conjugados


11


i 119911119911 = |119911|2

ii 1199111 + 1199112 = 1199111 + 1199112

iii 1199111 minus 1199112 = 1199111 minus 1199112

iv 1199111 times 1199112 = 1199111 times 1199112









119901 = (119886 + 119887119894) 120574



119901 = (119886 minus 119887119894)


1199012 = (119886 minus 119887119894) (119886 + 119887119894) 120574

= (1198862 + 1198872)|120574|2


119901 = 1198862 + 1198872



119901 = (119886 minus 119887119894) (119886 + 119887119894)

e 119873(119886 + 119894119887) = 119873(119886 minus 119894119887) lt 119873(119901) ∎



12














depende por sua vez



120572 = 120583120573 + 120588 119888119900119898 119873| 120588 | lt 119873| 120573 |






Em [29]

13




120572 = 120583120573 + 120588 119900119899119889119890 | 120588 | = 119889119894119904119905acirc119899119888119894119886 119886119900 119888119886119899119905119900 119898119886119894119904 119901119903oacute119909119894119898119900




2 119898119889119888 (120572 120573) = 120583120572 + 120584120573 119901119886119903119886 119886119897119892119906119899119904 120583 120584 isin 119885[119894]


120572 119900119906 120587 119889119894119907119894119889119890 120573











do

14

119901 = 1198862 + 1198872 = (119886 minus 119887119894)(119886 + 119887119894)


119901 = 119903119904 119888119900119898 1 lt 119903 119904 lt 119901 119890 119903 119904 isin 119885



15

CAPIacuteTULO 2

















matemaacutetico

















16


























nem lacetes




17





120595119866(119890) = 119906119907 119904119890 119890 119904oacute 119904119890 120595119867(120579(119890)) = 120593(119906)120593(119907)





120593 119881(119866) rarr 119881(119867) tal que

119906119907 isin 119864(119866) 119904119890 119890 119904oacute 119904119890 120593(119906)120593(119907) isin 119864(119867)





muacuteltiplas





18






Figura 4



Figura 5


Figura 6


19

Figura 7



Figura 8



20






adjacentes



Figura 11








Figura 12

21


119866 119888119886119904119900 119890119909119894119904119905119886 119888119886119904119900 119888119900119899119905119903aacute119903119894119900 119889119894119911119890119898119900119904 119902119906119890 119892(119866) = infin




diacircmetro




completo de ordem 5

Figura 13

119879119890119898119900119904 119889119894119886119898(119866) = 1 119903(119866) = 1 119892(119866) = 3 119890(119866) = 1



cruzam




grau par

22



23

CAPIacuteTULO 3











Exemplo 1198856 = 0 1 2 3 4 5

times 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

Figura 14 120591(1198856)

24























32 Exemplos

1198859 = 0 1 2 3 4 5 6 7 8

times 0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8

2 0 2 4 6 8 1 3 5 7

3 0 3 6 0 3 6 0 3 6

4 0 4 8 3 7 2 6 1 5

25

5 0 5 2 6 2 7 3 8 4

6 0 6 3 0 6 3 0 6 3

7 0 7 5 3 1 8 6 4 2

8 0 8 7 6 5 4 3 2 1

Figura 15 120591(1198859)








1198854 = 0 1 2 3

times 0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Figura 16 120591(1198854)

26




1198853 times 1198853 = ( ) isin 1198853=(0 0) (0 1) (0 2) (1 0) (1 1) (1 2) (2 0) (2 1) (2 2) ( ) ∙

( 119911) = (0 0)

(0 1) ∙ (1 0) = (0 0)

(0 2) ∙ (2 0) = (0 0)

Figura 17 120591(1198853 times 1198853)

11988525 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figura 18 120591(11988525)

1198852 times 1198654

1198852 = 0 1

1198751(119909) = 1199092 + 119909 + 1

1198752(119909) = 1199092 + 1

1198752(0) = 0 + 1 = 1

1198752(1) = 1 + 1 = 0

1198751(0) = 0 + 0 + 1 = 1

27

1198751(1) = 1 + 1 + 1 = 1

1198654 = 119886 + 119887119906 119886 isin 1198852 119890 119906 isin 1198654

119875(119906) = 1199062 + 119906 + 1

1198654 = 0 1 119906 119906 + 1

Tabela da soma

+ 0 1 119906 119906 + 1

0 0 1 119906 119906 + 1

1 1 0 119906 + 1 119906

119906 119906 119906 + 1 0 1

119906 + 1 119906 + 1 119906 1 0


times 0 1 119906 119906 + 1

0 0 0 0 0

1 0 1 119906 119906 + 1

119906 0 119906 119906 + 1 1

119906 + 1 0 119906 + 1 1 119906



1198852 times 1198654 = (119909 119910) 119909 isin 1198852 119890 119910 isin 1198654

(0 0) (0 1) (0 119906) (0 119906 + 1) (1 0) (1 1) (1 119906) (1 119906 + 1)

28

Figura 19 120591(1198852 times 1198654)



Figura 20











119868 onde 119868 = (1198831







29





1198852 times 1198857 =

(0 0) (0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (1 0) (1 1) (1 2) (1 3) (1 4) (1 5) (1 6)

Figura 21 120591(1198852 times 1198857)

1198853 times 1198855 =

(0 0) (0 1) (0 2 ) (0 3) (0 4) (1 0) (1 1) (1 2) (1 3) (1 4) (2 0) (2 1) (2 2) (2 3) (2 4)

Figura 22 120591(1198853 times 1198855)

30

















Em [36]

















119889119894119886119898 (120591(119877)) le 3 ∎


seja sobre ideais

31

Exemplos






de ideal principal













119892(120591(119877)) le 4 ∎












|120591(119877119899)| = 2119899 minus 1

32

Figura 23






119899 119890 119898119889119888(119909 119899) = 119889 (= 119880(119885119899) sub 119885(119885119899)lowast) e 119899119889 = |119881119889|
















33





times 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 4 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

34

Figura 24 120591(11988512)

35

CAPIacuteTULO 4


















dominante












36






1198852119899[119894] cong119885[119894]






0 119900119906 120572(2)119899minus1(1 + 1198941) = (2)119899minus1(1 + 1198941)









(1 + 1198941)2119899minus1




Teorema 4214 Para 119899 gt 1 119889119894119886119898(120591(1198852119899[119894])) = 2




1198941) 119901119900119903119905119886119899119905119900 119889119894119886119898(1198852119899[119894]) = 2 ∎

Teorema 4215 Para 119899 gt 1 119892(120591(1198852119899[119894])) = 3

37



Exemplo 119881(120591(1198854[119894])) = 2 1198942 2 + 1198942 1 + 1198941 1 + 1198943 3 + 1198941 3 + 1198943

Figura 25 120591(1198854[119894])






119881 (120591(119885119902119899[119894])) = langrang0







38





Teorema 4222 Para 119899 gt 1 119892 (120591(119885119902119899[119894])) = 3



119892 (120591(119885119902119899[119894])) = 3 ∎








119885[119894]




Exemplo 1198859 = 3 6 3119894 6119894 3 + 3119894 3 + 6119894 6 + 3119894 6 + 6119894

Figura 26 120591(1198859[119894])

39


119892 (120591(119885119901[119894])) = 4




(119885[119894]lang(119886 minus 119894119887)119899rang)




Teorema 4232 Para 119899 gt 1 119889119894119886119898 (120591(119885119901119899[119894])) = 3





( minus 119894)119899

hellip ( +

119894)119899

( minus 119894)119899minus1



Teorema 4233 Para 119899 gt 1 119892 (120591(119885119901119899[119894])) = 3








prod 119885119905119895

119899119895 [119894]119898119895=1 de tal modo que 120579( + 119894) = ((119909 119898119900119889(119905119895))

119899119895) + 119894(119910 119898119900119889(119905119895)

119899119895)119895=1

119898eacute um

isomorfismo


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)


40


119895=1 times prod 119901119904120573119904119897


prod (1199021198952120572119895 minus 119902119895

2120572119895minus2)119898119895=1 times prod (119901119904

120573119904 minus 119901119904120573119904minus1)

2119897119904=1 ) minus 1






1) 119889119894119886119898(120591(119885119899[119894])) = 1 se e somente se 119899 = 1199022








1) Se 119899119896 gt 1 119901119886119903119886 119886119897119892119906119898 119896 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3

2) Se 119899119896 = 1 119901119886119903119886 119905119900119889119900 119896 119890 119898 ge 3 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 3


4) Se 119899 = 1199021 times 1199022 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4

5) Se 119899 = 2 times 119902 119890119899119905atilde119900 119892(120591(119885119899[119894])) = 4


0 119895 ne 119896 e seja = (119895)

119895=1

119898isin

prod 119905119895119899119895119898


119892(120591(119885119899[119894])) = 3

2) Seja 119895 = 1 119895 = 1

0 119895 ne 1 119895 =

1 119895 = 2

0 119895 ne 2 119890 119911 =

1 119895 = 3

0 119895 ne 3 119890 119904119890119895119886 = (119895)

119895=1

119898 e = (119895)

119895=1

119898 e 119911 =

(119911)119895=1






41

Exemplo 1198855 = 2 + 1119894 2 + 4119894 3 + 1119894 3 + 4119894 1 + 2119894 1 + 3119894 4 + 2119894 4 + 3119894

Figura 27 120591(1198855[119894])



















42










jaacute que 11988511990211199022[119894] cong 1198851199021







seguinte teorema




119872cong 1198852 1198723 =

0 119890 |1198722| le 2










onde 119898 gt 1



43











120575(120591(119885119899[119894])) = 2



119896 (119888119895)

119895=1

119896

em 119881 (120591 (prod 119885119886119895

119899119895119896119895=1 )) Definimos 119909119895 =

1 119895 ne 119896

0 119895 = 119896 e assumindo que (119887119895)

119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896


119910119895 = 1 119895 ne 1

0 119895 = 1

Entatildeo (119887119895)119895=1

119896times (119910119895)

119895=1

119896ne (0)119895=1

119896 o que implica que (119888119895)119895=1

119896times (119909119895)

119895=1

119896= (0)119895=1

119896 Entatildeo 119888119895 =


1 119895 = 1

1 119895 = 119896

0 119900119906119905119903119900119904 119888119886119904119900119904

Entatildeo (119887119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1

119896 119890 (119888119895)119895=1

119896times (119911119895)

119895=1

119896ne (0)119895=1




( minus 119894)119898

119889119894119907119894119889119890 120572 119890 120573 = 0



( minus

119894)119898minus1

119889119894119907119894119889119890 120572 119901119900119903119905119886119899119905119900 119898 = ( + 119894)119898

( minus 119894)119898




∎


44


1198705 119900119906 11987033







|119885119902119898[119894]langrang| =1199022119898




22119898minus1 = 2 119890 |1198852119898[119894]| ge 64 assim 120591(119885119902119898[119894])


planar ∎



teorema


a) Se 1 le 119896 lt119899


b) Se 119899





119894)(119905119899 minus 119905119896)(1 + 1198941) = 0








45




2


2 119907 notin 119878 119890119899119902119906119886119899119905119900 119901119886119903119886

119899



119889119890119892(119907) = |119878| minus 1 1 le 119896 lt

119899

2

|119878| minus 2 119899

2le 119896 lt 119899

= 1199052119896 minus 1 1 le 119896 lt

119899

2

1199052119896 minus 2 119899

2le 119896 lt 119899

∎








eacute

Seja 119899 = prod 119905119895119899119895119898

119895=1 119888119900119898 119905119896 ne 119905119904 119901119886119903119886 119896 ne 119904 Definimos 119909119895 = 1 119895 = 119896

0 119895 ne 119896 e definimos 119910119895 =

1 119895 = 119904

0 119895 ne 119904

Seja = (119909119895) isin prod 119885119905119895119899119895

119898119895=1 [119894] 119890 = (119910119895) isin prod 119885119905119895

119899119895119898119895=1 [119894] Entatildeo 119889119890119892() =

119899

119905119896119899119896

ne119899

119905119904119899119904

minus 1 = 119889119890119892()







120591(119877) eacute par





46





2119899minus1(2119898 + 1 + 119894(2119896 + 1) ) = 2119899minus1(1 + 1198941) 119860119904119904119894119898 119889119890119892(1 + 1198941) = 1 ∎



se 119899 = 1 119890 119905119886119898119887eacute119898 119905 = 2 119900119906 119905 = 119901 equiv 1(1198981199001198894)


119899isin 119877 = prod 119877119895

119899119895=1 entatildeo 119860119899119899 ((119909119895)

119895=1

119899) =


119895=1

119899) = |119860119899119899 ((119909119895)

119895=1

119899)| minus


119895=1



















localmente empty

47



















Hamiltoniano














48


















119886119909 minus 119887119910 = 119902119898minus11198971

119887119909 + 119886119910 = 119902119898minus11198972












Hamiltoniano

49




∎











Hamiltoniano ∎






3) 120591(119885119899[119894]) eacute regular


5) 119899 = 119901 119900119906 119899 = 1199022


1) 119903119886119889(120591(119885119899[119894]) = 0 se e somente se 119899 = 2

2) 119903119886119889(120591(119885119899[119894]) = 1 se e somente se 119899 = 2119898 119900119906 119902119898 119900119899119889119890 119898 gt 1






( + 119894)119898

( minus 119894)119898minus1

( + 119894)119898minus1

( minus 119894)119898


50


( minus 119894)119898minus1


119894)119898minus1

( minus 119894)119898


119894)119898

( minus 119894)119898minus1


( minus 119894)119898


( minus 119894)119898minus1


( minus 119894)119898minus1


119903119886119889 (120591(119885119901119898[119894])) = 2 ∎









1198601 = (119909 0) 119909 isin 119885119905119898[119894] minus 0

1198602 = (0 119910) 119910 isin 119885119896[119894] minus 0

















∎

51


1 119898 gt 1 119888119900119898 119899 ne 119902 119901119886119903119886 119902119906119886119897119902119906119890119903 119902

119903119886119889(120591(119885119899[119894]) = 0 119899 = 2

1 119899 = 2119898119900119906 119902119898

2 119900119906119905119903119900 119888119886119904119900














par



par [1]





2le 119896 lt 119898



52


119895=1 119896 ge 2 119890 119895 = (119909119905) 119900119899119889119890

119909119905 = 1 119904119890 119905 = 1198950 119889119890 119900119906119905119903119886 119891119900119903119898119886


119899

(119886119895119898)


(11989921198981) minus 1





Euleriano












se


(ii) ∆(119866) le 4




53




















Teorema 4443

120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894])))

22119898minus1 minus 2 119904119890 119899 = 2119898 119898 ge 2

1199022119898minus2 minus 2 119904119890 119899 = 119902119898 119898 ge 2

1199012119898minus1 minus 1 119904119890 119899 = 119901119898 119898 ge 1


119895=1 prod 119902119895119897119895=1 prod 119902119895



Teorema 4444 119899 = 2119898 prod 119901119895119903119895119903

119895=1 prod 119902119895119897119895=1 prod 119902119895

119904119895119904119895=1 119900119899119889119890 119898 119903119895 ge 1 119890 119904119895 ge 2 entatildeo


119895=1 prod (1199021198952119904119895minus2 minus 1)119904

119895=1 minus 1

54


120596 (119871(120591(119885119899[119894]))) = 120594 (119871(120591(119885119899[119894]))) ∎

Diacircmetro de 119871(120591(119885119899[119894]))


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2 119902119906119886119899119889119900 119899 = 2119898 119900119906 119899 = 119902119898











1198872 119888 = 1199021198882 + 1198882 119890 119889 = 1199021198891 + 1198892 onde 0 lt 1198862 1198872 1198882 1198892 lt 119902 Assim


119886119889 + 119887119888 equiv 11988621198892 + 11988721198882 equiv 0(119898119900119889119902) (119868119868)



119902|1198862 119900119906 119902|(11988822 + 1198892

2) Visto que 1198862 lt 119902 119902|(11988822 + 1198892

2) Assim sendo 11988822 + 1198892

2 equiv 0(119898119900119889119902) e









55







11988911199021198961119894 11988621199021199032 + 11988721199021199042119894 11988821199021199052 + 11988921199021198962119894 ] isin 119881 (119871(120591(119885119899[119894]))) ∎


119889119894119886119898 (119871(120591(119885119899[119894]))) = 2



119885119899[119894] cong 1198851198991[119894] times 1198851198992

[119894]




119881 (119871(120591(119885119899[119894]))) = [(119906 119907) (0 119907)] 119906 119907 ne 0


∎


(ii) Se 119899 = 11990111198981199012

119897 onde 1199011 = 11988612 + 1198871

2 1199012 = 11988622 + 1198872

2 e 119898 119897 ge 1 entatildeo

119889119894119886119898 (119871(120591(119885119899[119894]))) = 3


119889119894119886119898 (119871(120591(119885119899[119894]))) = 3



119887119894)(119886 + 119887119894)) ] Entatildeo 119889(1199071 1199072) = 3

56






1198773 entatildeo 119889119894119886119898 (119871(120591(119877))) = 3





[(1198861 1198862 1198863) (1198881 1198882 1198883)] isin 119864 (119871(120591(119877))) Portanto 119889119894119886119898 (119871(120591(119877))) = 2


119889([(119909 0 1) (119910 1 0)] [(0 1 1) (1 0 0)]) = 3





119901 2119901 2119902 1199021 1199022 119902111990221199023 211990211199022 4119902 21199022 119901119902 119900119906 119899 = 2119898 119902119898 119888119900119898 119898 ge 2

(ii) 119889119894119886119898 (119871(120591(119885119899[119894]))) = 3 119889119890 119900119906119905119903119886 119891119900119903119898119886


Teorema 4451 119892 (119871(120591(119885119899[119894]))) = 3


119903119886119889 (119871(120591(119885119899[119894]))) = 2




57


119881 (119871(120591(119885119899[119894])))



todo [119909 119910] isin 119881 (119871(120591(119885119899[119894]))) ∎


119903119886119889 (119871(120591(119885119899[119894]))) = 2


[(119909 119910) (119905 119904)] isin 119881 (119871(120591(119885119899[119894])))







primo





Lema 4461 (i) 120574 (119871 (120591(119885119901[119894]))) = 120574119894´ (120591(119885119901[119894])) = 120574 ´ (120591(119885119901[119894])) = 119901 minus 1

(ii) 120574 (119871 (120591(11988511990211199022[119894]))) = 120574119894

´ (120591(11988511990211199022[119894])) = 120574 ´ (120591(11988511990211199022

[119894])) = 1199021 onde 1199021 lt 1199022



fato de que 120591(1198852119898[119894]) cong 120591(11988522119898[119894])

Teorema 4462 Para 119899 = 2119898 119898 ge 2

58

120574 (119871(120591(119885119899[119894]))) = 120574119894´(120591(119885119899[119894])) = 120574 ´(120591(119885119899[119894])) = lfloor

1












120574 (119871(120591(1198852119898[119894]))) = lfloor1




a ordem maacutexima



(i) 120596(120591(119885119899[119894])) = 2119898 minus 1

(ii) 120573(120591(119885119899[119894])) = 2119898(2119898 minus 1)







22119898minus119895 minus 1 ∎






59


119902119898 119898 ge 2

Lema 4466 (i) Para 119898 ge 2



119898



2rceilminus1



(1


1








arestas ∎






119889119890119892(119907) = 1199022119898119894119899119903119904 minus 2 119904119890 119903 119904 ge lceil

119898

2rceil

1199022119898119894119899119903119904 minus 1 119904119890 119903 119900119906 119904 lt lceil119898

2rceil


60







119901119898 119898 ge 2



Lema 44611 (i) Para 119898 ge 2



(ii) Para 119898 ge 3

1) Se 119861 = ⋃ ⋃ 119860119896119895119898minus1119895=119898minus119896







Se 1198821 = ⋃ ⋃ 119860119896119895119898minus119896minus1119895=1


119898minus119896minus1119896=1




120574 (119871(120591(119885119899[119894]))) = 120574 ´(120591(119885119899[119894])) = 120574119894´(120591(119885119899[119894])) = (1199042) + 119897 + 119887 se 119898 eacute par (1199042) + 119897 + 119887 + 1 se 119898

eacute iacutempar






61















outras arestas ∎

62

CAPIacuteTULO 5












grau de entrada











0 119890 |119886119899119899119897(119886) cup 119886119899119899119903(119886)| minus 2 119904119890 1198862 = 0






principais

63













Kaplansky









|119886119899119899119897(119906119886)| = |119886119899119899119897(119886119906)| = |119886119899119899119897(119886)|



119886119899119899119897(119886)




|119886119899119899119897(119886119906)| Isso prova o lema ∎



64


[3]




2 prod|119886119899119899(119889119894)|119899 minus 휀

119899

119894=1



119899

119894119895=1

119899

119894=1





matriz 119872119899(119877) temos

119883119863119860 = 0 harr 119909119894119895119889119895 = 0 119901119886119903119886 119894 119895 = 1 119899 (1)

119863119860119883 = 0 harr 119889119895119909119894119895 = 0 119901119886119903119886 119894 119894 = 1 hellip 119899 (2)


|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1


119899minus

휀 119900119899119889119890 휀 = 1 119886 119898119890119899119900119904 119902119906119890 1198602 = 0 119890 119899119890119904119905119890 119888119886119904119900 휀 = 2



|119886119899119899119897(119860)| cap |119886119899119899119903(119860)| = prod|119886119899119899(119889119894) cap 119886119899119899(119889119895)|

119899

119894119895=1

Assim sendo

65


119899

119894119895=1

119899

119894=1


119899119894=1 onde






temos

|119886119899119899119897(119860)| = |119886119899119899119903(119860)| = prod|119886119899119899(119889119894)|119899

119899

119894=1

= |119865|(119899minus119896)

e

|119886119899119899119897(119860)| cup |119886119899119899119903(119860)| = 2 prod|119865|119899(119899minus119896)

119899

119894=1

minus |119865|(119899minus119896)2






circuito Euleriano













66











119890119896)

























67




1 119886 119887

1 1 119886 119887

119886 119886 0 0

119887 119887 0 0



] no grafo 120591(1198722(119877)) o


∎

68

BIBLIOGRAFIA






296 (2) 462-479 (2006)


Algebra vol 274 314 (1) 168 (2004)


500-514 (1993)


434 (1999)


217434-447 (1999)




Reading MA (1969)








(1993)


Mold Mat 3 121-123 (2006)

69



671 (2007)







528 (1983)





215-218 (1964)


241-246 (1965)









Comm Algebra 37 (4) 1186 (2009)


(1949)

70



80 (1967)







(1995)














Prentice Hall(2006)




71






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