Some external characterizations of SV-rings and hereditary rings.
Let \(R\) be a ring, \(S\subset R\), and \(\text{Ann}_r(S)=\{a\in R\mid sa=0\) for all \(s\in S\}\). Then \(R\) is called a left AI-ring if every nonzero \(\text{Ann}_r(S)\) contains a nonzero idempotent, a (left) V-ring if every simple \(R\)-module is injective, a (left) semi-Artinian ring if every nonzero module has a nozero socle, and an SV-ring if \(R\) is a V-ring and a semi-Artinian ring. Then the authors characterize an SV-ring and a hereditary ring in terms of a left AI-ring, respectively. Theorem 1. The following statements are equivalent: (1) For every \(R\)-module \(M\), \(\text{End}_R(M)\) is left AI; (2) For every \(R\)-module \(M\) and every nonzero submodule \(N\) of \(M\), there exists an idempotent \(p\) of \(\text{End}_R(M)\) such that \(p(M)\subset N\); (3) Every nonzero \(R\)-module contains a nonzero injective submodule; and (4) \(R\) is a left SV-ring. Theorem~2. A ring \(R\) is left hereditary if and only if for every injective \(R\)-module \(M\), \(\text{End}_R(M)\) is right AI.
scientific article; zbMATH DE number 5282492