Multilinear polynomials and cocentralizing conditions in prime rings.
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 \(a\in U\), \(\alpha\in C\), or \(G(x)=cx+xc+\beta x\) for some \(c\in U\), \(\beta\in C\). 4. \(R\) satisfies the standard identity \(s_4\) and either \(F(x)=ax+xa+\alpha x\) for some \(a\in U\), \(\alpha\in C\), or \(G(x)=cx+xc+\beta x\) for some \(c\in U\), \(\beta\in C\).
scientific article; zbMATH DE number 6012535
scientific article