Locally injective modules and locally projective modules.
type: Item
instance of: scholarly article
title: Locally injective modules and locally projective modules.
zbMATH Open document ID: 1045.16003
author: Friedrich Kasch
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT32-4.html
review text: A module \(V\) is called locally injective if, for any non-essential submodule \(A\subseteq V\), there exists a nonzero injective submodule \(Q\subseteq V\) such that \(Q\cap A=0\). Dually, a module \(W\) is called locally projective if, for any non-small submodule \(B\subseteq W\), there exists a nonzero projective direct summand \(P\) of \(W\) such that \(P\subseteq B\). The author uses locally injective and locally projective modules to study the total \(\text{Tot}(M,W)\) of two modules \(M\) and \(W\). Here, \(\text{Tot}(M,W)\) is the set of all homomorphisms \(f\) in \(\Hom(M,W)\) such that the composition \(gf\) is an idempotent in \(\text{End}(M)\) for some \(g\) in \(\Hom(W,M)\). Several properties of these modules are obtained. In particular, a module is locally injective if and only if it is the injective envelope of a direct sum of injective modules.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 2012086
zbMATH Keywords: locally injective modules
zbMATH Keywords: locally projective modules
MaRDI profile type: Publication
OpenAlex ID: W2091723483
Wikidata QID: Q114059393
cites work: Q5485120
cites work: Q5485123
cites work: Q4836061
DOI: 10.1216/RMJM/1181070036
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description: scientific article; zbMATH DE number 2012086