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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the injective hulls of semisimple modules


Author: Jeffrey Levine
Journal: Trans. Amer. Math. Soc. 155 (1971), 115-126
MSC: Primary 16A52; Secondary 18G05
DOI: https://doi.org/10.1090/S0002-9947-1971-0306263-1
MathSciNet review: 0306263
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Abstract: Let $ R$ be a ring. Let $ T = { \oplus _{i \in I}}E(R/{M_i})$ and $ W = \prod\nolimits_{i \in I} {E(R/{M_i})} $, where each $ {M_i}$ is a maximal right ideal and $ E(A)$ is the injective hull of $ A$ for any $ R$-module $ A$. We show the following: If $ R$ is (von Neumann) regular, $ E(T) = T$ iff $ {\{ R/{M_i}\} _{i \in I}}$ contains only a finite number of nonisomorphic simple modules, each of which occurs only a finite number of times, or if it occurs an infinite number of times, it is finite dimensional over its endomorphism ring.

Let $ R$ be a ring such that every cyclic $ R$-module contains a simple. Let $ {\{ R/{M_i}\} _{i \in I}}$ be a family of pairwise nonisomorphic simples. Then $ E({ \oplus _{i \in I}}E(R/{M_i})) = \prod\nolimits_{i \in I} {E(R/{M_i})} $. In the commutative regular case these conditions are equivalent.

Let $ R$ be a commutative ring. Then every intersection of maximal ideals can be written as an irredundant intersection of maximal ideals iff every cyclic of the form $ R/\bigcap\nolimits_{i \in I} {{M_i}} $, where $ {\{ {M_i}\} _{i \in I}}$ is any collection of maximal ideals, contains a simple.

We finally look at the relationship between a regular ring $ R$ with central idempotents and the Zariski topology on spec $ R$.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0306263-1
Keywords: Regular, direct sum of modules, direct product of modules, injective, Zariski topology, semisimple
Article copyright: © Copyright 1971 American Mathematical Society