On a characterization of complete matrix rings
type: Item
instance of: scholarly article
title: On a characterization of complete matrix rings
zbMATH Open document ID: 0858.16022
DOI: 10.1007/BF02109794
author: L. van Wyk
published in: Periodica Mathematica Hungarica
publication date: 1997-01-22T00:00:00Z
review text: It was shown by P. R. Fuchs that a ring \(R\) is isomorphic to \(M_n(S)\) for some ring \(S\) and positive integer \(n\neq 1\) if and only if there are elements \(x\) and \(y\) of \(R\) such that (1) \(x^{n-1}\neq 0\), (2) \(x^n=0=y^2\), (3) \(x+y\) is a unit of \(R\), and (4) if \(r\) is an element of \(R\) with \(ryx^{n-1}=0\) then \(ry=0\). Fuchs also showed that (4) is redundant when \(n=2\). On the other hand, examples were constructed by Groenewald and the author to show that (4) does not follow from (1)-(3) if \(n\) is at least 3, but those examples were nevertheless \(n\) by \(n\) matrix rings. Here \(R\) is taken to be the direct sum of \(M_2(F)\) and \(M_n(F)\) for a field \(F\) and any positive integer \(n\) which is at least 3; there are elements \(x\) and \(y\) of \(R\) which satisfy conditions (1)-(3) but not (4); and it is easily shown that there is no positive integer \(k\neq 1\) and no ring \(S\) such that \(R\) is isomorphic to \(M_k(S)\).
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 938674
zbMATH Keywords: matrix rings
zbMATH Keywords: direct sums
reviewed by: A. W. Chatters
MaRDI profile type: Publication
cites work: Do isomorphic structural matrix rings have isomorphic graphs?
cites work: A characterisation result for matrix rings
cites work: On Rings for which Homogeneous Maps are Linear
cites work: A link between two characterisations of complete matrix rings
cites work: Recognition of matrix rings
cites work: Special radicals in structural matrix rings
full work available at URL: https://doi.org/10.1007/bf02109794
OpenAlex ID: W1979245019
Recommended article: A link between two characterisations of complete matrix rings
Recommended article: A characterisation result for matrix rings
Recommended article: A CHARACTERISATION OF MATRIX RINGS
Recommended article: Q4693650
Recommended article: ON THE REPRESENTATION OF FINITE RINGS BY MATRICES OVER COMMUTATIVE RINGS
Recommended article: Q3726290
Recommended article: On matrix rings over unit-regular rings
Recommended article: On some class of formal matrix ring
Recommended article: On matrix near-rings
Recommended article: Q3989553
description: scientific article; zbMATH DE number 938674