Elements in exchange rings with related comparability
type: Item
instance of: scholarly article
title: Elements in exchange rings with related comparability
zbMATH Open document ID: 0957.16005
DOI: 10.1155/S0161171200002234
author: Huanyin Chen
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-12-14T00:00:00Z
full work available at URL: https://eudml.org/doc/48664
review text: An associative unital ring \(R\) is said to satisfy related comparability provided that for any idempotents \(e\) and \(f\) in \(R\) such that \(e=1+ab\) and \(f=1+ba\) for some \(a,b\in R\), there exists a central idempotent \(u\) in \(R\) such that \(ueR\) is isomorphic to a direct summand of \(ufR\) and \((1-u)fR\) is isomorphic to a direct summand of \((1-u)eR\). In addition, \(R\) is said to be an exchange ring if for every right \(R\)-module \(A\) and any two decompositions \(A=M\oplus N=\bigoplus_{i\in I}A_i\), where \(M_R\cong R\) and the index set \(I\) is finite, then there exist submodules \(A_i'\subseteq A_i\) such that \(A=M\oplus(\bigoplus_{i\in I}A_i')\). The author shows that if \(R\) is an exchange ring, then the following are equivalent: (1) \(R\) satisfies related comparability; (2) Given \(a,b,d\in R\) with \(aR+bR=dR\), there exists a related unit \(u\in R\) such that \(a+bt=du\); (3) Given \(a,b\in R\) with \(aR=bR\), there exists a related unit \(u\in R\) such that \(a=bu\).
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 1496714
zbMATH Keywords: exchange rings
zbMATH Keywords: related comparability
zbMATH Keywords: related units
zbMATH Keywords: central idempotents
zbMATH Keywords: direct summands
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1496714