A Note on Posner's Theorem with Generalized Derivations on Lie Ideals

Abstract

Let $R$ be a prime ring of characteristic different from 2, $Z(R)$ its center, $U$ its Utumi quotient ring, $C$ its extended centroid, $G$ a non-zero generalized derivation of $R$, $L$ a non-central Lie ideal of $R$. We prove that if $[[G(u),u],G(u)]\in Z(R)$ for all $u\in L$ then one of the following holds:

1. there exists $\alpha \in C$ such that $G(x)=\alpha x$, for all $x\in R$;
2. satisfies the standard identity $s_4(x_1,\dots ,x_4)$ and there exist $a\in U$, $\alpha \in C$ such that $G(x)=ax+xa+\alpha x$, for all $x\in R$.