Identities with generalized derivations in prime rings (Q2120257)

Many authors characterize the structure of a prime ring \(R\) studying particular subsets of \(R\), involving the evaluations of some additive maps defined on \(R\). In this paper, the author considers the set \[P(F,G,H,S)=\biggl\{F(G(x)x)-H(x^2):x\in S\biggr\},\] where \(F,G\) and \(H\) are additive maps defined on \(R\) and \(S\subseteq R\). More precisely, he considers the case in which \(F,G\) and \(H\) are generalized derivations of \(R\) and \(S=f(R)\), the set of all evaluations of a non-central multilinear polynomial over \(C\) (the extended centroid of \(R\)): If \(\operatorname{char}(R)\neq2\) and \(P(F,G,H,f(R))=\{0\}\), then one of the following holds: \begin{itemize} \item[1.] \(F=H=0\); \item[2.] there exist \(a,b\in U\) such that \(F(x)=ax\), \(G(x)=bx\) and \(H(x)=abx\), for all \(x\in R\); \item[3.] there exist \(a,b\in U\), \(\lambda\in C\) such that \(F(x)=ax+xb\), \(G(x)=\lambda x\) and \(H(x)=\lambda(ax+xb)\), for all \(x\in R\); \item[4.] \(f(x_1,\ldots,x_n)\) is central-valued on \(R\) and \begin{itemize} \item[a)] either there exist \(a,b,c,p\in U\) such that \(F(x)=ax+xb\), \(G(x)=c x\) and \(H(x)=[p,x]+x(ac+cb)\), for all \(x\in R\); \item[b)] or there exist \(a\in U\) such that \(F=0\) and \(H(x)=[a,x]\), for all \(x\in R\). \end{itemize} \end{itemize}