Stable range one for exchange rings
type: Item
instance of: scholarly article
title: Stable range one for exchange rings
zbMATH Open document ID: 0837.16009
DOI: 10.1016/0022-4049(95)90029-2
author: Hua-Ping Yu
published in: Journal of Pure and Applied Algebra
publication date: 1996-04-24T00:00:00Z
review text: Let \(R\) be an associative ring with 1. \(R\) is called an exchange ring if for any two decompositions of a right \(R\)-module \(A\) of the form \(A\cong R\oplus N\cong\bigoplus_{i\in I}A_i\) there exist submodules \(A_i'\) of \(A_i\) such that \(A \cong R\oplus(\bigoplus_{i\in I}A_i')\). The author shows that exchange rings have stable range 1 under any of the following additional hypotheses: i) the primitive factors of \(R\) are Artinian; ii) the index of nilpotency of \(R\) is bounded; iii) the idempotents of \(R\) are central.
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 19B10
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 737485
zbMATH Keywords: decompositions
zbMATH Keywords: exchange rings
zbMATH Keywords: stable range 1
zbMATH Keywords: primitive factors
zbMATH Keywords: index of nilpotency
zbMATH Keywords: idempotents
reviewed by: Manfred Kolster
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)90029-2
OpenAlex ID: W2084715556
cites work: Rings all of Whose Pierce Stalks are Local
cites work: Exchange rings, units and idempotents
cites work: Homological properties of the ring of differential polynomials
cites work: Refinements for infinite direct decompositions of algebraic systems
cites work: Krull-Schmidt and cancellation over local rings
cites work: On the von Neumann regularity of rings with regular prime factor rings
cites work: Q4198895
cites work: Stable range one for rings with many units
cites work: Q3999650
cites work: On pi-regular rings whose primitive factor rings are Artinian
cites work: Q3995216
cites work: Lifting Idempotents and Exchange Rings
cites work: On rings whose projective modules have the exchange property
cites work: Bass's first stable range condition
cites work: Exchange rings and decompositions of modules
cites work: Cancellation of modules and groups and stable range of endomorphism rings
description: scientific article; zbMATH DE number 737485