AN ENGEL CONDITION WITH GENERALIZED DERIVATIONS ON LIE IDEALS

Abstract

Let $R$ be a prime ring, with extended centroid $C$, $g$ a non-zero generalized derivation of $R$, $L$ a non-central Lie ideal of $R$, $k\geq 1$ a fixed integer. If $[g(u),u]_k=0$, for all $u$, then either $g(x)=ax$, with $a \in C$ or $R$ satisfies the standard identity $s_4$. Moreover in the latter case either $char(R)=2$ or $char(R)\neq 2$ and $g(x)=ax+xb$ , with $a,b \in Q$ and $a-b\in C$. We also prove a more generalized version by replacing $L$ with the set $[I,I]$, where $I$ is a right ideal of $R$.