A note on an Engel condition with derivations in rings

Abstract

Let $R$ be a prime ring with center $Z(R)$, $C$ the extended centroid of $R$, $d$ a derivation of $R$ and $n,k$ be two fixed positive integers. In the present paper we investigate the behavior of a prime ring $R$ satisfying any one of the properties (i)~$d([x,y]_k)^n=[x,y]_k$ (ii) if $char(R)\neq 2$, $d([x,y]_k)-[x,y]_k\in Z(R)$ for all $x,y$ in some appropriate subset of $R$. Moreover, we also examine the case when $R$ is a semiprime ring