Lie ideals and Jordan derivations of prime rings

Abstract:

Herstein proved [1, Theorem 3.3] that any Jordan derivation of a prime ring of characteristic not 2 is a derivation of $R$. Our purpose is to extend this result on Lie ideals. We prove the following Theorem. Let $R$ be any prime ring such that char $R \ne 2$ ana let $U$ be a Lie ideal of $R$ such that ${u^2} \in U$ for all $u \in U$. If ,’, is an additive mapping of $R$ into itself satisfying $({u^2})’ = u’u + uu’$ for all $u \in U$, then $(u\upsilon )’ = u’\upsilon + u\upsilon ’$ for all $u,\upsilon \in U$.