Some characterizations of regular modules
type: Item
instance of: scholarly article
title: Some characterizations of regular modules
zbMATH Open document ID: 0722.16001
DOI: 10.5565/PUBLMAT_34290_02
author: Goro Azumaya
published in: Publicacions Matemàtiques
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41129
review text: Two characterizations of regularity in the sense of \textit{J. Zelmanowitz} [Trans. Am. Math. Soc. 163, 341-355 (1972; Zbl 0227.16022)] are proved. (A left R-module M is Zelmanowitz-regular provided that for each \(x\in M\) there exists a homomorphism \(h_ x: M\to R\) such that \(h_ x(x)x=x.)\) First, a module M is Zelmanowitz-regular if and only if every homomorphism f: \(L\to M\) is locally split in the sense that for each \(x\in f(L)\), there exists a homomorphism \(g_ x: M\to L\) such that \(fg_ x(x)=x\). Second, M is Zelmanowitz-regular if and only if (a) every cyclic submodule of M is a direct summand, and (b) M is locally projective in the sense that every epimorphism onto M is locally split.
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 4189942
zbMATH Keywords: regular module
zbMATH Keywords: locally projective module
zbMATH Keywords: flat strict Mittag-Leffler module
zbMATH Keywords: locally split homomorphism
zbMATH Keywords: characterizations of regularity
zbMATH Keywords: left R-module
zbMATH Keywords: Zelmanowitz-regular
zbMATH Keywords: cyclic submodule
zbMATH Keywords: direct summand
Wikidata QID: Q105666645
MaRDI profile type: Publication
OpenAlex ID: W1979930017
description: scientific article; zbMATH DE number 4189942