On a functional equation related to derivations in prime rings.
type: Item
instance of: scholarly article
title: On a functional equation related to derivations in prime rings.
zbMATH Open document ID: 1261.16042
author: Maja Fošner
author: Nina Peršin
published in: Monatshefte für Mathematik
publication date: 2012-11-21T00:00:00Z
review text: The authors consider a functional identity for a prime ring that leads to an extension of results about when a Jordan derivation of a prime ring is actually a derivation. Let \(R\) be a ring, \(D\colon R\to R\) a nonzero additive map, integers \(n,m\geq 1\), \(E(x)=2D(x^{m+n+1})-(m+n+1)(x^mD(x)x^n+x^nD(x)x^m)\), and \(p(X)=\sum x_{\sigma(1)}\cdots x_{\sigma(m+n+1)}\) taken over all \(\sigma\) in the symmetric group \(S_{m+n+1}\). The main result of the paper takes \(R\) to be a \(K\)-algebra, \(L\) a Lie subalgebra of \(R\) so that \(L\) is \(2(m+n+1)\)-free and \(p(L^{m+n+1})\subseteq L\). If \(E(x)=0\) for all \(x\in L\) then on \(L\), \(2D(x)=cx+\rho(x)\) for some \(c\in C(L)\) and \(\rho\colon L\to C(L)\). This result is used to prove that when \(R\) is a prime ring with \(\text{char\,}R=0\) or \(\text{char\,}R\geq m+n+1\) then \(E(r)=0\) for all \(r\in R\) forces \(R\) to be commutative and \(D\) to be a derivation of \(R\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16R60
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 6109333
zbMATH Keywords: functional identities
zbMATH Keywords: prime rings
zbMATH Keywords: derivations
zbMATH Keywords: additive maps
MaRDI profile type: Publication
full work available at URL: s00605-011-0319-z
OpenAlex ID: W2112219646
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DOI: 10.1007/S00605-011-0319-Z
description: scientific article; zbMATH DE number 6109333