Lie and Jordan structure in prime rings with derivations

Abstract

In this paper Lie ideals and Jordan ideals of a prime ring $R$ together with derivations on $R$ are studied. The following results are proved: Let $R$ be a prime ring, $U$ be a Lie ideal or a Jordan ideal of $R$ and $d$ be a nonzero derivation of $R$ such that $ud(u) - d(u)u$ is central in $R$ for all $u$ in $U$. (i) If the characteristic of $R$ is different from 2 and 3, then $U$ is central in $R$. (ii) If $R$ has characteristic 3 and $U$ is a Jordan ideal then $U$ is central in $R$; further, if $U$ is a Lie ideal with ${u^2} \in U$ for all $u$ in $U$, then $U$ is central in $R$. The case when $R$ has characteristic 2 is also studied. These results extend some due to Posner [2].