On Lie ideals with generalized derivations

Abstract

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a is an element of R and [a, f (U)] = 0 then a is an element of Z or d(a) = 0 or U subset of Z; (ii) If f(2)(U) = 0 then U subset of Z; (iii) If u(2) is an element of U for all u is an element of U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U subset of Z.