Torsion-freeness and non-singularity over right p.p.-rings.
type: Item
instance of: scholarly article
title: Torsion-freeness and non-singularity over right p.p.-rings.
zbMATH Open document ID: 1088.16017
author: John Dauns
author: Ulrich F. Albrecht
author: László Fuchs
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: Let \(R\) be an associative ring with identity, then a right \(R\)-module \(M\) is called non-singular if \(xK\neq 0\) for all non-zero elements \(x\in R\) and all essential right ideals \(K\) of \(R\). A right non-singular ring \(R\) is called right strongly non-singular if every finitely generated non-singular right \(R\)-module can be embedded in a free right \(R\)-module. A module \(M\) is called torsion-free if \(\text{Tor}_1^R(M,R/Rr)=0\) for all elements \(r\in R\). A ring \(R\) is right p.p. if every principal right ideal is projective, and \(R\) is right Utumi if \(R\) is right non-singular and every non-essential right ideal of \(R\) has a non-zero left annihilator. It is shown in the paper that, for a ring \(R\), the classes of torsion-free and non-singular right \(R\)-modules coincide if and only if \(R\) is a right Utumi p.p.-ring with no infinite set of orthogonal idempotents. The authors give several examples and applications of the main result. In particular, a characterization is given for right strongly non-singular rings \(R\) over which the classes of torsion-free and non-singular right \(R\)-modules coincide.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 2153335
zbMATH Keywords: torsion-free modules
zbMATH Keywords: non-singular modules
zbMATH Keywords: right p.p.-rings
zbMATH Keywords: right Utumi rings
zbMATH Keywords: principal right ideals
zbMATH Keywords: annihilators
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.10.020
OpenAlex ID: W1989838281
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DOI: 10.1016/J.JALGEBRA.2004.10.020
description: scientific article; zbMATH DE number 2153335