On an identity involving generalized derivations and Lie ideals of prime rings

Let R be a prime ring, U the Utumi quotient ring of R, C the extended centroid of R and L a noncentral Lie ideal of R. If R admits a generalized derivation F associated with a derivation [delta] of R such that for some fixed integers m, n [greater than or equal to] 1, F[([u, v]).sup.m] = [[u, v].sub.n] for all u, v [member of] L, then one of the following holds true: (i) R satisfies [s.sub.4], the standard identity in four variables. (ii) there exists [lambda] [member of] C such that F(x) = [lambda]x for all x [member of] R. Moreover, if n = 1, then [[lambda].sup.m] = 1 and if n > 1, then F = 0. Keywords: prime ring; generalized derivation; Lie ideal; GPIs Mathematics Subject Classification: 16W25, 16N60, 16R50