Separative cancellation for projective modules over exchange rings
type: Item
instance of: scholarly article
title: Separative cancellation for projective modules over exchange rings
zbMATH Open document ID: 0908.16002
DOI: 10.1007/BF02780325
author: Kenneth R. Goodearl
author: K. C. O'Meara
author: Pere Ara
author: Enrique Pardo Espino
published in: Israel Journal of Mathematics
publication date: 1999-03-16T00:00:00Z
review text: An \(R\)-module \(M\) has the exchange property if for every \(R\)-module \(A\) and any decompositions \(A=M'\oplus N=\bigoplus_{i\in I}A_i\), with \(M'\cong M\), there exist submodules \(A_i'\subseteq A_i\) such that \(A=M'\oplus\left(\bigoplus_{i\in I}A_i'\right)\). If the above condition is satisfied whenever the index set \(I\) is finite, \(M\) is said to satisfy the finite exchange property. A ring \(R\) is called an exchange ring if \(R_R\) satisfies the (finite) exchange property. A ring \(R\) is called a separative ring if the following condition holds for all \(A,B\in\text{FP}(R)\): \(A\oplus A\cong A\oplus B\cong B\oplus B\Rightarrow A\cong B\), where \(\text{FP}(R)\) is the class of all finitely generated projective modules over the ring \(R\). The separativity is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. In this paper, the authors develop some basic characterizations and initial applications of separativity, study stable rank conditions on exchange rings, prove the extension theorem for separative exchange rings, and give a corresponding extension result for the smaller class of strongly separative exchange rings. The results of this paper are particularly useful for the study of ring and module theory.
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 19B10
zbMATH DE Number: 1195644
zbMATH Keywords: decompositions
zbMATH Keywords: finite exchange property
zbMATH Keywords: separative rings
zbMATH Keywords: finitely generated projective modules
zbMATH Keywords: cancellation problems
zbMATH Keywords: exchange rings
zbMATH Keywords: separativity
zbMATH Keywords: stable rank conditions
reviewed by: Jule Zhang
MaRDI profile type: Publication
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Separative cancellation for projective modules over exchange rings (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf02780325

Separative cancellation for projective modules over exchange rings (OpenAlex ID)
OpenAlex ID: W1972067889