Characterization of rings using direct-projective modules and direct- injective modules
type: Item
instance of: scholarly article
title: Characterization of rings using direct-projective modules and direct- injective modules
zbMATH Open document ID: 0788.16007
DOI: 10.1016/0022-4049(93)90073-3
author: Xue Weimin
published in: Journal of Pure and Applied Algebra
publication date: 1994-06-07T00:00:00Z
review text: A module \(M\) is said to be direct-projective if, given any summand \(N\) of \(M\) with projection \(p:M \to N\) and any epimorphism \(f: M\to N\), there is a \(g\in \text{End}_ R(M)\) such that \(gf = p\). Similarly, a module \(M\) is called direct-injective if given any summand \(N\) of \(M\) with injection \(i: N\to M\) and any monomorphism \(f: N\to M\), there exists a \(g\in \text{End}_ R(M)\) such that \(fg = i\). The author uses direct-projective and direct-injective modules to give new characterizations of hereditary, semihereditary and semisimple rings. For example, he shows that the following are equivalent for any ring \(R\): (1) \(R\) is left hereditary. (2) Every submodule of a projective \(R\)-module is direct-projective. (3) Every principal left ideal of \(S = \text{End}_ R(F)\) is direct- projective for any free \(R\)-module \(F\). (4) Every factor module of an injective \(R\)-module is direct-injective. (5) Every sum of two injective submodules of an \(R\)-module is direct-injective. (6) Every sum of two isomorphic injective submodules of an \(R\)-module is direct-injective.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 219106
zbMATH Keywords: left hereditary ring
zbMATH Keywords: summand
zbMATH Keywords: direct-injective modules
zbMATH Keywords: semisimple rings
zbMATH Keywords: projective \(R\)-module
zbMATH Keywords: direct-projective
zbMATH Keywords: injective submodules
Wikidata QID: Q114215367
reviewed by: Paul E. Bland
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90073-3
OpenAlex ID: W2027688439
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description: scientific article; zbMATH DE number 219106