A RESULT ON GENERALIZED DERIVATIONS IN PRIME RINGS

Let R be a prime ring, H a generalized derivation of R, L a noncentral Lie ideal of R, and 0 ≠ a ∈ R. Suppose that au^s(H(u))^nu^t = 0 for all u ∈ L, where s, t ≥ 0 and n 0 are fixed integers. If s = 0, then H(x) = bx for all x ∈ R, where b ∈ U, the right Utumi quotient ring of R, with ab = 0 unless R satisfies s4, the standard identity in four variables. If s 0, then H = 0 unless R satisfies s4.