On a theorem of J. Vukman
type: Item
instance of: scholarly article
title: On a theorem of J. Vukman
zbMATH Open document ID: 0858.16031
DOI: 10.1007/BF01818330
author: Motoshi Hongan
published in: Aequationes Mathematicae
publication date: 1997-03-06T00:00:00Z
full work available at URL: https://eudml.org/doc/137658
review text: Let \(R\) be a ring. A biadditive map \(D:R\times R\to R\) is called a biderivation if the maps \(x\mapsto D(x,y)\) and \(x\mapsto D(y,x)\) are derivations for each \(y\in R\). A biderivation \(D\) is said to be symmetric if \(D(x,y)=D(y,x)\) for all \(x,y\in R\). The main result of the paper states that if \(R\) is a noncommutative prime ring of characteristic different from 2, \(I\) is a nonzero ideal of \(R\), and \(D\) is a symmetric biderivation of \(R\) such that \([D(x,x),x]\) is central for every \(x\in I\), then \(D=0\). Reviewer's remark. The reviewer proved [J. Algebra 172, No. 3, 764-786 (1995; Zbl 0827.16024)] that every biderivation \(D:I\times I\to R\), where \(R\) is a noncommutative prime ring and \(I\) is its ideal, is of the form \(D(x,y)=\lambda[x,y]\) where \(\lambda\) is an element of the extended centroid of \(R\). In particular, this implies that actually every symmetric biderivation of \(I\times I\) into \(R\) is zero, unless \(R\) has characteristic 2.
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 935131
zbMATH Keywords: biadditive maps
zbMATH Keywords: derivations
zbMATH Keywords: prime rings
zbMATH Keywords: symmetric biderivations
zbMATH Keywords: extended centroid
reviewed by: Matej Brešar
MaRDI profile type: Publication
cites work: Derivations in Prime Rings
cites work: Symmetric bi-derivations on prime and semi-prime rings
OpenAlex ID: W2072631462
description: scientific article; zbMATH DE number 935131