Jordan derivations and antiderivations on triangular matrices
type: Item
instance of: scholarly article
title: Jordan derivations and antiderivations on triangular matrices
zbMATH Open document ID: 1072.15021
author: Dominik Benkovič
published in: Linear Algebra and its Applications
publication date: 2005-05-04T00:00:00Z
review text: Let \(\mathcal{C}\) be a commutative ring with 1, \(\mathcal{T}_{n}\) be the ring of all upper triangular matrices over \(\mathcal{C}\) and \(\mathcal{M}\) be a \(\mathcal{T}_{n}\)-bimodule. Then a \(\mathcal{C}\)-linear map \(\Delta :\mathcal{T}_{n}\rightarrow\mathcal{M}\) is a Jordan derivation if \(\Delta(ab+ba)=\Delta(a)b+a\Delta(b)+\Delta(b)a+b\Delta(a)\) for all \(a,b\). The author proves that every Jordan deriviation \(\Delta\) is uniquely expressible as the sum of a derivation and an antiderivation (that is, a \(\mathcal{C} \)-linear map \(\delta\) such that \(\delta(ab)=\delta(b)a+b\delta(a)\)) which vanishes on the diagonal matrices. No such decompositon exists for the free algebra \(\mathcal{C}\left\langle x,y\right\rangle \) in noncommuting variables \(x,y\). As a corollary, the author deduces that, for \(n\geq2\), every Jordan derivation of \(\mathcal{T}_{n}\) into itself is a derivation.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 17A15
zbMATH DE Number: 2165359
zbMATH Keywords: triangular matrix algebra
zbMATH Keywords: Jordan derivations
zbMATH Keywords: antiderivation
MaRDI profile type: Publication
full work available at URL: j.laa.2004.10.017
OpenAlex ID: W1964937073
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DOI: 10.1016/J.LAA.2004.10.017
description: scientific article; zbMATH DE number 2165359