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Advisor(s)
Abstract(s)
O nosso principal objectivo é estudar o trabalho de G. Ehrlich e também de R. Andruszkiewicz e E.R. Puczylowski no que concerne aos anéis liais, isto é, anéis em que a relação de ideal é transitiva. Começamos por resumir algum material de base que será usado ao longo do trabalho. Em particular, referimo-nos a anéis primos e semiprimos, ao Radical de Jacobson e algumas das suas propriedades; apresentamos a construção de um anel de fracções a partir de um anel comutativo e registamos algumas propriedades da estrutura dos seus ideais. Além disso, recordamos algumas de nições básicas e conceitos acerca dos domínios de ideais principais e dos domínios de factorização única. Em seguida, de nimos anel lial e apresentamos vários exemplos. São dadas várias caracterizações de anéis liais e, em particular, de anéis liais semiprimos, Artinianos e nilpotentes. Também é investigada a lialidade de domínios de integridade. Por último, é apresentada uma ligação entre os grupos abelianos e os anéis liais.
Our main objective is to study the work of G. Ehrlich and also of R. Andruszkiewicz and E.R. Puczylowski concerning lial rings; that is, rings in which the ideal relation is transitive. We begin by summarizing some basic material which will be used in the sequel. In particular, we refer to prime and semiprime rings, to the Jacobson radical and to some of their properties; we present the construction of a ring of fractions of a commutative ring and note some properties of its ideal structure. In addition, we recall some basic de nitions and concepts concerning principal ideal domains and unique factorization domains. Next, we de ne a lial ring and present various examples. Several characterizations of lial rings are given and, in particular, of semiprime, Artinian and nilpotent lial rings. Filiality of integral domains is also investigated. Finally, a link between abelian groups and lial rings is presented.
Our main objective is to study the work of G. Ehrlich and also of R. Andruszkiewicz and E.R. Puczylowski concerning lial rings; that is, rings in which the ideal relation is transitive. We begin by summarizing some basic material which will be used in the sequel. In particular, we refer to prime and semiprime rings, to the Jacobson radical and to some of their properties; we present the construction of a ring of fractions of a commutative ring and note some properties of its ideal structure. In addition, we recall some basic de nitions and concepts concerning principal ideal domains and unique factorization domains. Next, we de ne a lial ring and present various examples. Several characterizations of lial rings are given and, in particular, of semiprime, Artinian and nilpotent lial rings. Filiality of integral domains is also investigated. Finally, a link between abelian groups and lial rings is presented.
Description
Keywords
Anéis (Matemática) Grupos abelianos Anel filial
Citation
Publisher
Universidade da Beira Interior