Lie and Jordan structure in prime rings with derivations

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 u² ∈ 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].