Non-induced isomorphisms of matrix rings
type: Item
instance of: scholarly article
title: Non-induced isomorphisms of matrix rings
zbMATH Open document ID: 0888.16015
DOI: 10.1007/BF02760690
published in: Israel Journal of Mathematics
publication date: 1998-06-03T00:00:00Z
review text: If a ring \(R\) has the property that there are distinct positive integers \(i\) and \(j\) such that the free \(R\)-modules of ranks \(i\) and \(j\) are isomorphic, then the corresponding matrix rings \(M_i(R)\) and \(M_j(R)\) are isomorphic. The author shows that it is possible for \(M_i(R)\) and \(M_j(R)\) to be isomorphic even when the corresponding free modules are not isomorphic. In particular, a ring \(R\) is constructed with the properties that \(R\) is isomorphic to \(M_2(R)\) but \(R\) is not isomorphic as a module to the direct sum of two copies of itself. In a note added in proof, an outline is given of a subsequent construction due to Bergman which yields a ring \(R\) such that \(M_i(R)\) and \(M_j(R)\) are isomorphic for all positive integers \(i\) and \(j\), but the corresponding free modules are isomorphic if and only if \(i=j\).
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 1062239
zbMATH Keywords: free modules
zbMATH Keywords: matrix rings
author: Gene Abrams
reviewed by: A. W. Chatters
MaRDI profile type: Publication
cites work: Q3744314
cites work: The Module Type of a Ring
full work available at URL: https://doi.org/10.1007/bf02760690
OpenAlex ID: W1994229400
description: scientific article; zbMATH DE number 1062239