Certain differential identities on prime rings and Banach algebras (Q6597483)

Several authors, starting from a well-known result of \N\textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093--1100 (1958; Zbl 0082.03003)], have studied suitable identities involving additive maps defined on (semi)prime rings. Recalling that, for a prime ring \(R\), an additive map \(d:R\longrightarrow R\) is a \textit{derivation} if \(d(xy)=d(x)y+xd(y)\), for all \(x,y\in R\) and \(F:R\longrightarrow R\) is called \textit{generalized derivation} if \(F(xy)=F(x)y+xd(y)\), for all \(x,y\in R\), where \(d\) is a derivation of \(R\), here the authors study a particular identity involving three different generalized derivations with the aim of describing the structure of the ring or the form of the maps.\N\NIn particular they consider \(L\) a Lie ideal of a prime ring \(R\) with characteristic different from \(2\) and \(F_1,F_2,F_3\) three generalized derivations of \(R\) satisfying the following identity: \N\[\NF_1(x)\bot F_2(y)=F_3(x)\bot y\N\]\Nfor all \(x,y\in L\), where \(\bot\) represents either the Lie product \([., .]\), or the Jordan product \(\circ\). In both cases they prove that one of the following holds:\N\begin{itemize}\N\item[1.] \(L\subseteq Z(R)\);\N\item[2.] \(F_1=F_3=0\);\N\item[3.] there exists \(\lambda\in C\) (the extended centroid of \(R\)) such that \(F_3(x)=\lambda F_1(x)\) and \(F_2(x)=\lambda x\), for all \(x\in R\);\N\item[4.] \(R\subseteq M_2(C)\), the \(2 \times 2\) matrix ring over \(C\).\N\end{itemize}\N\NFurthermore, as an application, the same identity is studied locally on nonvoid open subsets of prime Banach algebras.