AN ENGEL CONDITION OF GENERALIZED DERIVATIONS WITH ANNIHILATOR ON LIE IDEAL IN PRIME RINGS

Let R be a prime ring with its Utumi ring of quotients U, C - Z (U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k >= 2 a fixed integer. Suppose that there exists 0 not equal a is an element of R such that a[F(u(n1)), u(n2), ..., u(nk)] = 0 for all u is an element of L, where n(1), n(2), ..., n(k) >= 1 are fixed integers. Then either there exists lambda is an element of C such that F(x) = lambda x for all x is an element of R, or R satisfies s(4), the standard identity in four variables.