Completion of diagrams by automorphisms and Bass' first stable range condition
type: Item
instance of: scholarly article
title: Completion of diagrams by automorphisms and Bass' first stable range condition
zbMATH Open document ID: 0839.16007
author: M. J. Canfell
published in: Journal of Algebra
publication date: 1996-05-28T00:00:00Z
review text: Weak projectivity and injectivity conditions for a module \(P\) are studied and related to the stable range of the endomorphism ring of \(P\). For instance, \(P\) is defined to be quasi-projective (respectively, epi-projective) if for any epimorphism \(g:P\to M\) and any homomorphism (respectively, epimorphism) \(f:P\to M\), there exists a homomorphism (respectively, epimorphism) \(h:P\to P\) such that \(f=gh\). One of the main theorems of the paper gives a number of equivalent conditions for the endomorphism ring of a quasi-projective module \(P\) to have stable range 1; in particular, this occurs if and only if \(P\) is epi-projective. Another main theorem enlarges the list of equivalent conditions by specializing to free modules of rank 1. For example, a ring \(R\) has stable range 1 if and only if every cyclic right \(R\)-module \(M\) is uniquely generated, meaning that whether \(M= xR=yR\), there exists a unit \(u\in R\) such that \(x=yu\). The paper also contains a number of general results on epi-projectivity and unique generation, as well as a discussion of the dual problems for quasi-injective modules.
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 19B10
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 815921
zbMATH Keywords: stable range
zbMATH Keywords: endomorphism rings
zbMATH Keywords: quasi-projective modules
zbMATH Keywords: free modules
zbMATH Keywords: cyclic right modules
zbMATH Keywords: epi-projectivity
zbMATH Keywords: unique generation
zbMATH Keywords: quasi-injective modules
MaRDI profile type: Publication
full work available at URL: jabr.1995.1255
OpenAlex ID: W2076375096
DOI: 10.1006/JABR.1995.1255
description: scientific article; zbMATH DE number 815921