Modules whose hereditary pretorsion classes are closed under products.
type: Item
instance of: scholarly article
title: Modules whose hereditary pretorsion classes are closed under products.
zbMATH Open document ID: 1121.16028
author: John E. van den Berg
author: Robert Wisbauer
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-12T00:00:00Z
review text: Let \(R\) be a ring with identity. It was shown by \textit{J. A. Beachy} and \textit{W. D. Blair} [Commun. Algebra 6, 1-34 (1978; Zbl 0405.16005)] that the following conditions on a ring \(R\) with identity are equivalent: (1) Every hereditary pretorsion class in \(R\)-Mod is closed under arbitrary direct products. (2) Every left \(R\)-module \(M\) is finitely annihilated, that is, \((0:M)=(0:X)\) for some finite subset \(X\) of \(M\). (3) \(R\) is left Artinian. In this paper, the authors attempt to describe those modules \(M\) with the property that every hereditary pretorsion class in \(\sigma[M]\) is closed under products in \(\sigma[M]\). Such a module \(M\) is called product closed. Then, as is shown by the authors, if \(M\) is finitely generated product closed, and is projective in \(\sigma[M]\), then every hereditary pretorsion class in \(\sigma[M]\) is subgenerated by an \(M\)-generated module. This is one of the authors' main results and extends Beachy and Blair's characterization of left Artinian rings.
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5124469
zbMATH Keywords: product closed modules
zbMATH Keywords: finitely annihilated modules
zbMATH Keywords: left Artinian rings
zbMATH Keywords: hereditary pretorsion classes
zbMATH Keywords: categories of left modules
zbMATH Keywords: direct products
zbMATH Keywords: Artinian modules
reviewed by: Yoshiki Kurata
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.05.030
OpenAlex ID: W2022604474
cites work: On Quasi‐Artinian Rings
cites work: Finitely annihilated modules and orders in artinian rings
cites work: Q3941569
cites work: Q4723914
cites work: Zur Bewertungstheorie nicht kommutativer Körper
cites work: Q4049223
cites work: When every torsion preradical is a torsion radical
cites work: Q5422326
cites work: Every algebraic chain is the congruence lattice of a ring
cites work: On rings (and chain domains) with restricted completeness conditions on topologizing filters
cites work: Rings whose kernel functors are linearly ordered
cites work: Module categories with linearly ordered closed subcategories
cites work: Q4002547
cites work: Q4881597
DOI: 10.1016/J.JPAA.2006.05.030
description: scientific article; zbMATH DE number 5124469