Commutativity of $\ast$-Prime Rings with Generalized Derivations

Abstract

Let $R$ be a 2-torsion free $\ast$-prime ring and $F$ be a generalized derivation of $R$ with associated derivation $d$. If $U$ is a $\ast$-Lie ideal of $R$ then in the present paper, we shall show that $U\subseteq Z(R)$ if $R$ admits a generalized derivation $F$ (with associated derivation $d$) satisfying any one of the properties: (i) $F[u,v]=[F(u),v]$, (ii) $F(u\circ v)=F(u)\circ v$, (iii) $F[u,v]=[F(u),v]+[d(v),u]$, (iv) $F(u\circ v)=F(u)\circ v+d(v)\circ u$, (v) $F(uv)\pm uv=0$ and (vi) $d(u)F(v)\pm uv=0$ for all $u,v\in U$.