On the injective hulls of semisimple modules

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 $R$.