Multilinear polynomials and cocentralizing conditions in prime rings.
type: Item
instance of: scholarly article
title: Multilinear polynomials and cocentralizing conditions in prime rings.
zbMATH Open document ID: 1243.16047
DOI: 10.1134/S0037446611050065
author: Vincenzo De Filippis
author: Francesco Rania
published in: Siberian Mathematical Journal
publication date: 2012-03-07T00:00:00Z
review text: Assume \(R\) is a noncommutative prime ring and denote by \(Z(R)\) its center and by \(U\) the Utumi ring of quotients of \(R\). Denote further the extended centroid of \(R\) by \(C\), that is \(C=Z(U)\). Posner established that for a nontrivial derivation \(d\) of \(R\), the condition \([[d(x_1),x_1],x_2]=0\) for all \(x_i\in R\) implies the commutativity of \(R\). In case \(\text{char\,}R\neq 2\) Vukman showed that \([[d(x_1),x_1],x_1]=0\) implies commutativity for \(R\). Lanski obtained that if the Posner condition above holds for each \(x_2\in R\) and each \(x_1\) in a noncommutative Lie ideal of \(R\) then either \(R\) is commutative or \(R\) is of characteristic different from 2 and satisfies the standard identity of degree 4. Beidar, Brešar and Chebotar studied functional identities in prime rings and obtained the following result. Let \(d\) be a derivation and \(F\) an additive map on \(R\) such that \([d(x),F(x)]=0\) for each \(x\in R\). Assuming \(\text{char\,}R\neq 2\) they proved that \(F(x)=\lambda d(x)+\mu(x)\) where \(\lambda\in C\) and \(\mu\colon R\to C\) is additive. A map \(G\colon R\to R\) is a generalized derivation if \(G(xy)=G(x)y+xd(y)\) for all \(x,y\in R\), where \(d\) is a derivation of \(R\). For a multilinear polynomial in the free associative algebra over \(C\), \(f(x_1,\dots,x_n)\), denote by \(f(R)\) the set \(\{f(r_1,\dots,r_n)\mid r_i\in R\}\). Assume that the above \(f\) is not a central polynomial for \(R\), and suppose \(F\) and \(G\) are two generalized derivations of \(R\) satisfying, for all \(x\) and \(y\in f(R)\), the condition \([[F(x),x],[G(y),y]]=0\). The main result of the paper under review is that under the above conditions, assuming that \(\text{char\,}R\neq 2\), one of the following four statements holds. 1. \(F(x)=\alpha x\) for some \(\alpha\in C\). 2. \(G(x)=\beta x\) for some \(\beta\in C\). 3. \(f^2\) assumes values only in the centre of \(R\), and either \(F(x)=ax+xa+\alpha x\) for some \(a\in U\), \(\alpha\in …
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16R60
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 6012535
zbMATH Keywords: prime rings
zbMATH Keywords: differential identities
zbMATH Keywords: functional identities
zbMATH Keywords: generalized derivations
zbMATH Keywords: commutativity theorems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6012535