Commuting maps on some subsets of matrices that are not closed under addition
type: Item
instance of: scholarly article
title: Commuting maps on some subsets of matrices that are not closed under addition
zbMATH Open document ID: 1247.15026
author: Willian Franca
published in: Linear Algebra and its Applications
publication date: 2012-05-14T00:00:00Z
review text: Let \(M_{n}(\mathbb K)\) be the ring of all \(n\times n\) matrices over a field \(\mathbb K\). The author describes all additive mappings \(G:M_{n}(\mathbb K)\to M_{n}(\mathbb K)\) satisfying the condition \(G(x)x=xG(x)\) for all \(x\) from one of the following subsets of \(M_{n}(\mathbb K)\): (i) the set of all singular \(x\in M_{n}(\mathbb K)\), and (ii) the set of all invertible \(x\in M_{n}(\mathbb K)\); in the second case the extra assumption \(|\mathbb K|>2\) is made. In both cases such a \(G\) has the form \(x\mapsto\lambda x + \mu(x)\), where \(\lambda\) is a scalar multiple of the identity matrix and \(\mu\) is an additive mapping of \(M_{n}(\mathbb K)\) into its centre. An illustrative example shows that in (ii) the assumption on \(|\mathbb K|\) must not be dropped.
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 6034537
zbMATH Keywords: commuting maps
zbMATH Keywords: invertible matrices
zbMATH Keywords: singular matrices
zbMATH Keywords: linear preserver problems
zbMATH Keywords: matrix rings
MaRDI profile type: Publication
full work available at URL: j.laa.2012.02.018
OpenAlex ID: W2032217258
cites work: Centralizing mappings and derivations in prime rings
cites work: Commuting maps: a survey.
cites work: Q3417002
cites work: Linear Preserver Problems
DOI: 10.1016/J.LAA.2012.02.018
description: scientific article; zbMATH DE number 6034537