On semi-artinian \(V\)-modules
type: Item
instance of: scholarly article
title: On semi-artinian \(V\)-modules
zbMATH Open document ID: 0786.16002
DOI: 10.1016/0022-4049(92)90008-4
author: Nguyen Viet Dung
author: Patrick F. Smith
published in: Journal of Pure and Applied Algebra
publication date: 1993-04-01T00:00:00Z
review text: The starting point of this paper is the fact that, as a consequence of a theorem of \textit{B. L. Osofsky} and \textit{P. F. Smith} [J. Algebra 139, 342-354 (1991; Zbl 0737.16001)], a right \(R\)-module \(M\) (over an arbitrary ring \(R\)) is semisimple if and only if every cyclic subquotient of \(M\) is \(M\)-injective. Observing that semisimple modules are semi-artinian (i.e., every nonzero image of \(M\) has nonzero socle) and are \(V\)-modules (i.e., every simple module is \(M\)-injective), the authors extend this fact in a significant way by proving that, if \(M\) is quasi-projective, then it is a semi-artinian \(V\)-module if and only if every nonzero cyclic subquotient of \(M\) contains a nonzero \(M\)-injective module. The paper ends by giving a method for producing examples of semi-artinian \(V\)-modules of prescribed socle length.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 149932
zbMATH Keywords: semisimple modules
zbMATH Keywords: \(V\)-modules
zbMATH Keywords: simple module
zbMATH Keywords: \(M\)-injective
zbMATH Keywords: quasi-projective modules
zbMATH Keywords: cyclic subquotient
zbMATH Keywords: semi-artinian \(V\)-modules
zbMATH Keywords: socle length
reviewed by: José L. Gómez-Pardo
MaRDI profile type: Publication
full work available at URL: 0022-4049(92)90008-4
OpenAlex ID: W1966457160
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description: scientific article; zbMATH DE number 149932