Recognition and computations of matrix rings
type: Item
instance of: scholarly article
title: Recognition and computations of matrix rings
zbMATH Open document ID: 0883.16020
DOI: 10.1007/BF02937312
author: André Leroy
author: T. Y. Lam
published in: Israel Journal of Mathematics
publication date: 1998-03-15T00:00:00Z
review text: This paper continues the study of criteria for a ring \(R\) to be a full \(n\) by \(n\) matrix ring \(M_n(A)\) over some ring \(A\) and methods for constructing such a base-ring \(A\) (which is not in general unique up to isomorphism). For instance it was shown by Agnarsson, Amitsur and Robson that if \(m\) and \(n\) are positive integers then \(R\) is isomorphic to \(M_{m+n}(A)\) for some ring \(A\) if and only if \(R\) has elements \(a,b,f\) such that \(f^{m+n}=0\) and \(af^m+f^nb=1\); it is shown here that, when this happens, \(A\) can be taken to be the eigenring \(E_R(fR)\) which, by definition, is \(I(fR)/fR\) where \(I(fR)\) is the idealiser of the right ideal \(fR\) of \(R\). These results are used to show that certain (factor rings) of Ore extensions are matrix rings and to construct corresponding base-rings as eigenrings. Also, many new criteria are given for \(R\) to be a full \(n\) by \(n\) matrix ring, several of them involving the existence of an \(n\)-th root for a matrix over \(R\). For example, one criterion is that there exists a positive integer \(r\neq q\) such that the \(r\) by \(r\) matrix with 1's immediately below the main diagonal and 0's elsewhere should have an \(n\)-th root in \(M_r(R)\). Finally, the authors extend a result of Levy, Robson and Stafford on ``hidden matrices'' by showing that if \(H\) is the ring of integer quaternions and \(c\) is a positive integer, then \(H+M_2(cH)\) is a full 2 by 2 matrix ring if and only if all the prime factors of \(c\) are congruent to 1 modulo 4.
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 997595
zbMATH Keywords: existence of roots
zbMATH Keywords: idealisers
zbMATH Keywords: Ore extensions
zbMATH Keywords: matrix rings
zbMATH Keywords: base rings
zbMATH Keywords: eigenrings
reviewed by: A. W. Chatters
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02937312
OpenAlex ID: W2063140365
description: scientific article; zbMATH DE number 997595