A note on an Engel condition with generalized derivations in rings

Abstract

Let $R$ be a prime ring with characteristic different from two, $I$ a nonzero ideal of $R$ and $F$ a generalized derivation associated with a nonzero derivation $d$ of $R$. In the present paper we investigate the commutativity of $R$ satisfying the relation $F([x,y]_k)^n=([x,y]_k)^l$ for all $x,y\in I$, where $l,n, k$ are fixed positive integers. Moreover, let $R$ be a semiprime ring, $A=O(R)$ be an orthogonal completion of $R$ and $B=B(C)$ the Boolean ring of $C$. Suppose $F([x,y]_k)^n=([x,y]_k)^l$ for all $x,y\in R$, then there exists a central idempotent element $e$ of $B$ such that $d$ vanishes identically on $eA$ and the ring $(1-e)A$ is commutative.