Vincenzo De Filippis and Shuliang Huang DI.S.I.A., Faculty of Engineering, Chuzhou University

Abstract : Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $n$ a fixed positive integer. If $R$ admits a generalized derivation $F$ associated with a derivation $d$ such that $(F([x,y]))^{n}=[x,y]$ for all $x,y\in I$. Then either $R$ is commutative or $n=1$, $d=0$ and $F$ is the identity map on $R$. Moreover in case $R$ is a semiprime ring and $(F([x,y]))^{n}=[x,y]$ for all $x,y\in R$, then either $R$ is commutative or $n=1$, $d(R)\subseteq Z(R)$, $R$ contains a non-zero central ideal and $F(x)-x \in Z(R)$ for all $x\in R$.

Keywords : prime and semiprime rings, differential identities, generalized derivations