Rings with internal cancellation.
type: Item
instance of: scholarly article
title: Rings with internal cancellation.
zbMATH Open document ID: 1076.16004
author: Dinesh Khurana
author: T. Y. Lam
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: The paper is an exhaustive study of the class of rings that satisfy internal cancellation with respect to their one-sided ideals. The authors propose the name of IC rings for such rings and give various new characterizations of them in terms of suitable stable range conditions as well as unique generator properties of idempotent generated right ideals. A ring \(R\) is said to be of regular stable range 1 if, for any regular elements \(a,b\in R\), \(aR+bR=R\) implies that \(a+bx\) is a unit of \(R\). An element \(a\in R\) is said to have the right unique generator property if, for any \(b\in R\), \(aR=bR\) implies that \(b=au\) for some unit \(u\) of \(R\). Then it is shown that a ring is IC if and only if it has regular stable range 1 if and only if all idempotents (or equivalently, all regular (resp. unit-regular) elements) have the right unique generator property. The authors study the functorial behavior of IC rings and characterize the rings whose matrix rings are IC as the rings whose finitely generated projective modules satisfy cancellation. Two elements \(a,b\in R\) are called similar if \(a=u^{-1}bu\) for some unit \(u\) of \(R\), and pseudo-similar if there exist \(x,y,z,w\in R\) such that \(a=zbx\), \(b=xaw\) and \(x=xzx=xwx\). It is shown that a regular element \(x=xyx\in R\) is unit-regular if and only if there exists a unit \(u\) of \(R\) such that \(xy=xu\) and \(yx=ux\). The authors use this result to study pseudo-similarity and prove that IC rings are precisely those rings for which similarity and pseudo-similarity are the same. The paper concludes with a treatment of ideals for which idempotents lift modulo all one-sided subideals and with an appendix by R. G. Swan on the failure of cancellation for finitely generated projective modules over complex group algebras.
reviewed by: Septimiu Crivei
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 19B10
zbMATH DE Number: 2136971
zbMATH Keywords: regular stable range 1
zbMATH Keywords: unique generator property
zbMATH Keywords: pseudo-similarities
zbMATH Keywords: lifting idempotents
zbMATH Keywords: internal direct sum cancellation
zbMATH Keywords: regular elements
zbMATH Keywords: stable range conditions
zbMATH Keywords: idempotent generated right ideals
zbMATH Keywords: exchange rings
zbMATH Keywords: finitely generated projective modules
zbMATH Keywords: unit-regular elements
zbMATH Keywords: complex group algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.07.032
OpenAlex ID: W2045244266
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DOI: 10.1016/J.JALGEBRA.2004.07.032
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