Some external characterizations of SV-rings and hereditary rings.
type: Item
instance of: scholarly article
title: Some external characterizations of SV-rings and hereditary rings.
zbMATH Open document ID: 1152.16005
DOI: 10.1155/2007/84840
author: H. Rahnaoui
author: A. Haily
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/54863
review text: 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.
reviewed by: George Szeto
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16P20
zbMATH DE Number: 5282492
zbMATH Keywords: AI-rings
zbMATH Keywords: endomorphism rings
zbMATH Keywords: semi-Artinian V-rings
zbMATH Keywords: hereditary rings
zbMATH Keywords: simple modules
MaRDI profile type: Publication
OpenAlex ID: W1977911153
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description: scientific article; zbMATH DE number 5282492