On right ideals and derivations in prime rings with Engel condition

Abstract

Let $R$ be an associative ring with center $Z(R)$ and $d$ a nonzeroderivation of $R$. The main object in this paper is to study thesituation $[[d(x^r)x^n,x^r{]}_s,[y,d(y){]}_t{]}^mınZ(R)$ for all $x,y$ in some appropriate subset of $R$, where $n\ge 0$, $s\ge 0$, $t\ge 0$, $m\ge 1$, $r\ge 1$ arefixed integers and $R$ is a prime or semiprime ring.