An Engel condition of generalized derivations with annihilator on Lie ideal in prime rings

Abstract

Let $R$ be a prime ring with itsUtumi ring of quotients $U$, $C=Z(U)$ extended centroid of $R$,$F$ a nonzero generalized derivation of $R$, $L$ a noncentralLie ideal of $R$ and $k\geq 2$ a fixed integer. Suppose that there exists $0\neq aın R$ suchthat $a[F(u^{n_1}),u^{n_2},\ldots,u^{n_k}]=0$for all $u ın L$, where $n_1, n_2, \ldots, n_k\geq 1$ are fixedintegers. Then either there exists $\lambdaın C$ such that $F(x)=\lambdax$ for all $xın R$, or $R$ satisfies $s_4$, the standardidentity in four variables.