On some classes of semi-Artinian rings.
type: Item
instance of: scholarly article
title: On some classes of semi-Artinian rings.
zbMATH Open document ID: 1276.16001
DOI: 10.1134/S0037446612050011
author: A. N. Abyzov
published in: Siberian Mathematical Journal
publication date: 2013-01-31T00:00:00Z
review text: As it is admitted, this article is a continuation of the author's work [in [1] = Sib. Mat. Zh. 49, No. 4, 721-738 (2008); translation in Sib. Math. J. 49, No. 4, 575-586 (2008; Zbl 1154.16004); Sib. Mat. Zh. 50, No. 3, 481-488 (2009); translation in Sib. Math. J. 50, No. 3, 379-384 (2009; Zbl 1212.16019)]. A right \(R\)-module \(M\) is called weakly regular if any submodule of \(M\) not in the Jacobson radical of \(M\), has a nonzero direct summand of \(M\). If all right \(R\)-modules are weakly regular there may exist a left \(R\)-module which is not weakly regular, [see [1, loc. cit.] Corollaries 3.8, 3.9, and their remark]. An associative ring \(R\) with \(1\neq 0\), not necessarily commutative, is called a right semiartinian ring (resp. \(V\)-ring) if every non-zero right module over \(R\) has a simple submodule (resp. if the Jacobson radical of every right module over \(R\) is zero). By an \(SV\)-ring one means a ring which is both right and left semiartinian \(V\)-ring. In the main result of this article (Theorem 3.11), which describes the rings over which all modules (right or left) are weakly regular, the author gives seven equivalent conditions for a ring \(R\) to be an \(SV\)-ring. Let us mention some of the interesting ones [which are also in [1, loc. cit.], Theorem 3.7]: \(R\) is an \(SV\)-ring if and only if \(R\) is a regular (in the sense of von Neumann) ring over which all modules (whether right or left) are weakly regular, or if and only if \(R\) is a right \(V\)-ring with the latter property. Concerning the latter two equivalences, it is interesting to recall that in commutative rings, \(V\)-rings and regular rings coincide, whereas in non-commutative rings this coincidence may not occur. A ring \(R\) is called mod-retractable if for any right \(R\)-module \(M\) and any nonzero submodule \(N\) of \(M\), \(\Hom(M,N)\neq 0\). \textit{A. Haghany, M. R. Vedadi} and the reviewer [in Bull. Iran. Math. Soc. 35, No. 2, 37-45 (2009; Zbl 1194.16007)] have sh…
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 6133170
zbMATH Keywords: weakly regular modules
zbMATH Keywords: quasiprojective modules
zbMATH Keywords: semi-Artinian rings
zbMATH Keywords: SV-rings
zbMATH Keywords: CSL-rings
zbMATH Keywords: direct summands
zbMATH Keywords: retractable rings
zbMATH Keywords: retractable modules
zbMATH Keywords: right duo rings
MaRDI profile type: Publication
cites work: Q3545603
cites work: Q5391566
cites work: Rings over which all modules are \(I_0\)-modules. II.
cites work: Rings over which all modules are \(I_0\)-modules.
cites work: Q4002547
cites work: Q4200205
cites work: Projective injective modules
cites work: CORETRACTABLE MODULES
cites work: Q3057374
cites work: Regular Endomorphism Rings of Free Modules
cites work: An analogue of the pierce sheaf for non-commutative rings
cites work: Asymmetry in the Converse of Schur's Lemma
cites work: Perfect rings for which the converse of Schur's lemma holds
cites work: Q4001907
cites work: On semi-artinian \(V\)-modules
cites work: Rings whose modules have maximal submodules
cites work: On quasi-duo rings
cites work: Semiartinian \(V\)-rings and semiartinian von Neumann regular rings
cites work: Q4009825
full work available at URL: s0037446612050011
OpenAlex ID: W2091834112
description: scientific article; zbMATH DE number 6133170