Generalized derivations with central values on lie ideals LIE IDEALS

Abstract

let R be a prime ring of charR = 2, H a generalized derivation and L a noncentral lie ideal of R. We show that if l s H(l)l t ∈ Z(R) for all l ∈ L, where s, t ≥ 0 are fixed integers, then H(x) = bx for some b ∈ C, the extended centroid of R, or R satisfies S 4 . Moreover, let R be a 2-torsion free semiprime ring, let A = O(R) be an orthogonal completion of R andwhere s, t ≥ 0 are fixed integers. Then there exists idempotent e ∈ B such that H(x) = bx on eA and the ring (1 − e)A satisfies S 4 .