Generalized derivations and left multipliers on Lie ideals.
type: Item
instance of: scholarly article
title: Generalized derivations and left multipliers on Lie ideals.
zbMATH Open document ID: 1223.16017
author: Basudeb Dhara
author: Vincenzo De Filippis
author: Rajendra Kumar Sharma
published in: Aequationes Mathematicae
publication date: 2011-06-17T00:00:00Z
review text: Let \(R\) be a prime ring with \(\text{char\,}R\neq 2\), extended centroid \(C\), Utumi quotient ring \(U\), noncentral Lie ideal \(L\), and generalized derivations \(F\) and \(D\). For fixed positive integers \(n\) and \(m\) let \(P(x)\) be the expression \(F(x^{n+m+1})-F(x)x^{n+m}-x^nF(x)x^m\). The main result in the paper assumes that \(P(x)\) is an identity for \(L\) and shows that either \(D=0\) and \(F(r)=ar\) for some \(a\in U\) and all \(r\in R\), or else \(R\) satisfies the standard polynomial identity \(s_4\) of degree four. The first possibility must hold when \(P(x)\) is an identity for \(R\). When \(R\) satisfies \(s_4\) and \(P(x)\) is an identity for \(L\), then \(n+m\) even forces \(D=0\), and \(n+m\) odd forces \(D\) to be a derivation and there is \(a\in U\) so that for all \(r\in R\), \(F(r)=ar-D(r)\) when \(n\) is even and \(F(r)=ar+D(r)\) when \(n\) is odd.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 5909092
zbMATH Keywords: prime rings
zbMATH Keywords: generalized derivations
zbMATH Keywords: Lie ideals
zbMATH Keywords: identities with derivation
MaRDI profile type: Publication
full work available at URL: s00010-011-0082-1
OpenAlex ID: W1975169860
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DOI: 10.1007/S00010-011-0082-1
description: scientific article; zbMATH DE number 5909092