On Lie ideals and Jordan left derivations of prime rings

Abstract

summary:Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that $u^{2} \in U$ for all $u \in U$. In the present paper it is shown that if $d$ is an additive mappings of $R$ into itself satisfying $d(u^{2})=2ud(u)$ for all $u \in U$, then $d(uv)=ud(v)+vd(u)$ for all $u,v \in U$.