Annihilator condition of a pair of derivations in prime and semiprime rings

Abstract

Let n be a fixed positive integer, R be a prime ring, D and G two derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(D(u)un−unG(u)) = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then one of the following holds: 1. D = G = 0, unless R satisfies s4; 2. char (R) ≠ 2, R satisfies s4, n is even and D = G; 3. char (R) ≠ 2, R satisfies s4, n is odd and D and G are two inner derivations induced by b, c respectively such that b + c ∈ C; 4. char (R) = 2 and R satisfies s4.