Exchange rings and their extensions
type: Item
instance of: scholarly article
title: Exchange rings and their extensions
zbMATH Open document ID: 1023.16012
DOI: 10.1016/S0022-4049(02)00299-2
author: Chan Yong Hong
author: Nam Kyun Kim
author: Yang Lee
published in: Journal of Pure and Applied Algebra
publication date: 2003-04-09T00:00:00Z
review text: An exchange ring is a ring \(R\) for which the module \(R_R\) (equivalently, \(_RR\)) has the finite exchange property [cf. \textit{R. B. Warfield}, jun., Math. Ann. 199, 31-36 (1972; Zbl 0228.16012)]. The authors first study the (von Neumann) regularity of such a ring. Their main theorem states that if \(R\) is an exchange ring whose primitive factor rings are all Artinian, and if all homomorphic images of \(R\) are semiprimitive, then \(R\) is regular -- in fact, unit-regular. This extends a result of \textit{H.-P. Yu} [Commun. Algebra 25, No. 2, 661-670 (1997; Zbl 0873.16007)], who had proved that under the above hypotheses, \(R\) is strongly \(\pi\)-regular. Applications concerning strong regularity and strong \(\pi\)-regularity follow from the main theorem. The second section of the paper is concerned with transferring the exchange property from a ring \(R\) to related rings. For example, the authors prove that if \(\alpha\) is a ring endomorphism of \(R\), then \(R\) is an exchange ring if and only if the skew power series ring \(Rx;\alpha\) is an exchange ring.
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16L30
zbMATH DE Number: 1895985
zbMATH Keywords: exchange rings
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: strongly \(\pi\)-regular rings
zbMATH Keywords: skew power series rings
MaRDI profile type: Publication
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full work available at URL: s0022-4049(02)00299-2
OpenAlex ID: W2027118642
description: scientific article; zbMATH DE number 1895985