A note on multiplicative(generalized)-derivations and Lie ideals in prime and semiprime rings

Abstract

Let $R$ be a 2-torsion free semiprime ring and $U$ be a square closed Lie ideal of $R$. A mapping $F: R \longrightarrow R$ is called a multiplicative (generalized)- derivation if there exists a map $d: R \longrightarrow R$ such that $F(x y)=F(x) y+x d(y)$, for all $x, y \in R$. Suppose that $R$ admits a multiplicative(generalized)-derivation $F$ associated with a map $d$ such that $d(U) \subseteq U$. In the present paper, we shall prove that $d$ is commuting on $U$ if one of the following conditions holds: (i) $F([x, y])= \pm[d(x), y],(i i) F(x o y)= \pm(d(x) o y),(i i i) F([x, y])= \pm(d(x) o y),(i v) F(x o y)= \pm[d(x) o y],(v) F([x, y])=$ $\pm[F(x), y],($ vi) $F($ xoy $)= \pm[F(x)$ oy $],($ vii $) F([x, y])= \pm[F(x)$ oy $],($ viii $) F($ xoy $)= \pm[F(x), y]$ for all $x, y \in U$.