Endomorphism algebras of modules with distinguished torsion-free elements
type: Item
instance of: scholarly article
title: Endomorphism algebras of modules with distinguished torsion-free elements
zbMATH Open document ID: 0859.13004
author: Steve T. Files
published in: Journal of Algebra
publication date: 1997-04-10T00:00:00Z
review text: Let \(E(M)\) denote the algebra of \(R\)-linear endomorphisms of a unitary \(R\)-module \(M\), \(R\) being a fixed discrete valuation ring. The author contributes to the complex of problems, which arose from Kaplansky's result, that every isomorphism \(E(M)\to E(N)\) is induced by an isomorphism of the modules themselves if both \(M\) and \(N\) are torsion. He introduces the concept of a stable element \(x\) of a mixed module \(M\), that is an element for which (i) there exists \(\delta\in E(M)\) such that \(\delta(x)=x\) and \(\delta M/\langle x\rangle\) is torsion, and (ii) \(M/E(M)x\) is torsion. The two main results then read as follows: (1) Assume \(M\) and \(N\) are reduced modules and the torsion of \(M\) is totally projective. If each module possesses a stable element, then the implication of Kaplansky's result holds. (2) This is true also if only \(M\) possesses a stable element under the additional condition \(\text{rk} (N)\leq \text{rk} (M)\), and both being finite.
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13C12
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 826749
zbMATH Keywords: endomorphisms
zbMATH Keywords: discrete valuation ring
zbMATH Keywords: torsion
zbMATH Keywords: stable element
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: jabr.1995.1348
OpenAlex ID: W2013613932
DOI: 10.1006/JABR.1995.1348
description: scientific article; zbMATH DE number 826749