Stable range one for rings with many units
type: Item
instance of: scholarly article
title: Stable range one for rings with many units
zbMATH Open document ID: 0653.16013
DOI: 10.1016/0022-4049(88)90034-5
author: Kenneth R. Goodearl
author: Pere Menal
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Here is the first stable range condition for a ring R with 1 (introduced by H. Bass): If a, b are in R and \(aR+bR=R\), then there is y in R such that \(R(a+by)=R\). In a stronger condition called in the paper (right) unit 1-stable range condition, one requires the above y to be a unit of R. Here is an even stronger condition: for any a, b in R there is a unit u of R such that a-u and \(b-u^{-1}\) are both units. This condition is verified in the paper in a number of interesting cases. For any unital complex C *-algebra R, the above 3 conditions are equivalent.
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 4065201
zbMATH Keywords: strongly \(\pi \) -regular von Neumann regular rings
zbMATH Keywords: first stable range condition
zbMATH Keywords: unit 1-stable range condition
zbMATH Keywords: units
zbMATH Keywords: unital complex C *- algebra
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90034-5
OpenAlex ID: W2054468949
cites work: On injective and surjective endomorphisms of finitely generated modules
cites work: Q5831378
cites work: Q3723166
cites work: Q5185036
cites work: Q4106702
cites work: Einreihig zerlegbare Ringe sind selbstdual
cites work: Krull-Schmidt and cancellation over local rings
cites work: Q3708935
cites work: Power-cancellation of groups and modules
cites work: Q4198895
cites work: Cancellation of low-rank vector bundles
cites work: Surjective endomorphisms of finitely generated modules
cites work: Q4748637
cites work: Stable Range in AW ∗ Algebras
cites work: On a Theorem of Flanders
cites work: Generators and relations for certain linear groups over rings of linear operators
cites work: The stable range of \(C^*\)-algebras
cites work: On pi-regular rings whose primitive factor rings are Artinian
cites work: Q3795816
cites work: \(K_ 1\) of von Neumann regular rings
cites work: Stable range in C*-algebras
cites work: Vector Bundles and Projective Modules
cites work: Regular rings with involution
cites work: Stable rank of rings and dimensionality of topological spaces
cites work: Bass's first stable range condition
cites work: Q3737598
description: scientific article; zbMATH DE number 4065201