\(\sigma^*\)-projective covers and \(\sigma\)-Artinian rings
type: Item
instance of: scholarly article
title: \(\sigma^*\)-projective covers and \(\sigma\)-Artinian rings
zbMATH Open document ID: 0594.16018
DOI: 10.1016/0022-4049(86)90055-1
author: A. C. Benander
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let R be a ring with unit element, and let \(\sigma\) be a hereditary torsion theory of right R-modules. If \(P\to^{\alpha}Q\to C\to 0\) is an exact sequence with C \(\sigma\)-torsion, then \(\alpha\) is minimal if and only if Cok(M\(\to P\to^{\alpha}Q)\) is \(\sigma\)-torsion implies that Cok\((M\to P)\) is \(\sigma\)-torsion. A module P is \(\sigma^*\)-projective if every \(\sigma\)-dense submodule of P is \(\sigma\)-projective. If P and M are modules, then P is a \(\sigma^*\)-projective cover of M if P is \(\sigma^*\)-projective and there exists an exact sequence \(P\to^{\alpha}M\to C\to 0\) such that \(\alpha\) is minimal and C is \(\sigma\)-torsion. Some basic properties of \(\sigma^*\)-covers are given. If R has the descending chain condition on \(\sigma\)-closed right ideals, then every \(\sigma\)-cocritical module has a \(\sigma^*\)-projective cover. If R has the descending chain condition on \(\sigma\)-closed right ideals, then \(\sigma\) is a perfect torsion theory if and only if, for any injective cogenerator E associated with \(\sigma\), \(_{\Lambda}E\) is quasi-injective over its endomorphism ring \(\Lambda\).
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16Exx
zbMATH DE Number: 3955149
zbMATH Keywords: hereditary torsion theory
zbMATH Keywords: exact sequence
zbMATH Keywords: \(\sigma^*\)-projective cover
zbMATH Keywords: \(\sigma^*\)-covers
zbMATH Keywords: descending chain condition
zbMATH Keywords: closed right ideals
zbMATH Keywords: cocritical module
zbMATH Keywords: perfect torsion theory
zbMATH Keywords: injective cogenerator
reviewed by: Mark L. Teply
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90055-1
OpenAlex ID: W2000083097
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cites work: Q5642687
description: scientific article; zbMATH DE number 3955149