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 $k\ge 2$ a fixed integer. Suppose that there exists $0\ne aınR$ suchthat $a[F(u^{n_1}),u^{n_2},\dots ,u^{n_k}]=0$for all $uınL$, where $n_1,n_2,\dots ,n_k\ge 1$ are fixedintegers. Then either there exists $\lambda ınC$ such that $F(x)=\text{lambdax}$ for all $xınR$, or $R$ satisfies $s_4$, the standardidentity in four variables.