Generalized derivations with central values on lie ideals LIE IDEALS

Let R be a prime ring of H a generalized derivation and L a noncentral lie ideal of R. We show that if l^sH(l)l^t in Z(R) for all lin2 L, where s, t> 0 are fixed integers, then H(x) = bx for some b in C, the extended centroid of R, or R satisfies S4. Moreover, let R be a 2-torsion free semiprime ring, let A = O(R) be an orthogonal completion of R and B = B(C) the Boolean ring of C. Suppose ([x1; x2]sH([x1; x2])[x1; x2]t in Z(R) for all x1; x2 in R, where s, t> 0 are fixed integers. Then there exists idempotent e in B such that H(x) = bx on eA and the ring (1-e)A satisfies S4.