Characterizations of semiperfect and perfect rings
type: Item
instance of: scholarly article
title: Characterizations of semiperfect and perfect rings
zbMATH Open document ID: 0863.16014
DOI: 10.5565/PUBLMAT_40196_08
author: Xue Weimin
published in: Publicacions Matemàtiques
publication date: 1996-09-18T00:00:00Z
full work available at URL: https://eudml.org/doc/41249
review text: Adapting the notion of a generalized projective cover introduced by \textit{G. Azymaya} [Ring Theory, Proc. Biennial Ohio State-Denison Conf., May 1992, 28-40 (1993; Zbl 0854.16014)], the author defines a module epimorphism \(f:P\to M\) to be a (generalized) cover for \(M\) if \(\text{Ker}(f)\) (\(\text{Ker}(f)\subseteq\text{Rad}(P))\) is superfluous in \(P\). If here \(P\) is also projective then the (generalized) cover is said to be projective and likewise if \(P\) is locally projective. \((P\) is called locally projective if, given any module epimorphism \(g:A\to B\) and any homomorphism \(f:P\to B\), then for every finitely generated submodule \(P_0\) of \(P\) there is a homomorphism \(h:P\to A\) such that \(f=gh\) on \(P_0\).) The paper has three sections. The first gives some basic properties of the concepts just defined. The second uses (generalized) projective covers to give characterizations of semiperfect modules and various generalizations of these. In the final section he uses (generalized) locally projective covers to characterise left perfect, semiperfect and left quasi-perfect rings. (The concept of a quasi-perfect ring was introduced \textit{V. Camillo} and the author [in Commun. Algebra 19, No. 10, 2841-2850 (1991; Zbl 0742.16009)].) In particular he shows that the following statements are equivalent for a ring \(R\): (1) \(R\) is left perfect (semiperfect, left quasi-perfect, resp.); (2) every (finitely generated, artinian resp.) left \(R\)-module has a locally projective cover; (3) every (finitely generated, artinian resp.) left \(R\)-module has a generalized locally projective cover.
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 911929
zbMATH Keywords: generalized projective covers
zbMATH Keywords: module epimorphisms
zbMATH Keywords: finitely generated submodules
zbMATH Keywords: semiperfect modules
zbMATH Keywords: locally projective covers
zbMATH Keywords: left quasi-perfect rings
zbMATH Keywords: generalized locally projective covers
reviewed by: John Clark
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 911929