Cancellation of projective modules over regular rings with comparability
type: Item
instance of: scholarly article
title: Cancellation of projective modules over regular rings with comparability
zbMATH Open document ID: 0851.16008
DOI: 10.1016/0022-4049(95)00034-8
author: Pere Ara
author: K. C. O'Meara
author: D. V. Tyukavkin
published in: Journal of Pure and Applied Algebra
publication date: 1996-11-24T00:00:00Z
review text: Let \(R\) be a regular ring, and let \(s\) and \(t\) be positive integers. \(R\) satisfies \((s:t)\)-comparability if for any \(x,y\in R\), either \(t(xR)\lesssim s(yR)\) or \(t(yR)\lesssim s(xR)\); \(R\) satisfies \(s\)-comparability if for any \(x,y \in R\), either \(xR\lesssim s(yR)\) or \(yR\lesssim s(xR)\). It is clear that \((s:t)\)-comparability implies \(s\)-comparability. Open Problem 4 in \textit{K. R. Goodearl}'s book Von Neumann regular rings [Pitman, London (1979; Zbl 0411.16007), Krieger, Malabar, Florida, 2nd ed. (1991; Zbl 0749.16001)]: If \(R\) is a directly finite regular ring which satisfies \((s:t)\)-comparability for some integers \(s \geq t > 0\), is \(R\) unit-regular? This paper answers, in the negative, this problem: a directly finite regular ring with \(s\)-comparability is constructed which is not unit-regular. Moreover, new results on directly finite regular rings with \(s\)-comparability are given. In particular, the authors prove the following result: Let \(R\) be a directly finite regular ring satisfying \(s\)-comparability for some positive integer \(s\). Then for all finitely generated projective right \(R\)-modules \(A\), \(B\), \(C\), \(A\oplus C\cong B\oplus C\) and \(C\lesssim nA\) for some \(n\in\mathbb{N}\Rightarrow A\cong B\).
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 896030
zbMATH Keywords: unit-regular rings
zbMATH Keywords: stable range
zbMATH Keywords: \((s:t)\)-comparability
zbMATH Keywords: \(s\)-comparability
zbMATH Keywords: directly finite regular rings
zbMATH Keywords: finitely generated projective right modules
reviewed by: Jule Zhang
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00034-8
OpenAlex ID: W2060660845
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Wikidata QID: Q126437424
description: scientific article; zbMATH DE number 896030