Notes on (?, ?)-derivations of lie ideals in prime rings

Abstract

Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that u2 ? U for all u ? U and d be a nonzero (?, ?)-derivation of R. We prove the following results: (i) If [d(u), u]?, ? = 0 or [d(u), u]?, ? ? C?, ? for all u 2 U, then U ? Z. (ii) If a 2 R and [d(u), a]?, ? = 0 for all u ? U, then U ? Z or a ? Z. (iii) If d([u, v]) = ±[u, v]?, ? for all u 2 U, then U ? Z. © 2012 The Korean Mathematical Society.