Generalized derivations with left annihilator conditions in prime and semiprime rings

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) be the extended centroid of R, H and G two generalized derivations of R, L a noncentral Lie ideal of R, I a nonzero ideal of R. The left annihilator of S ⊆ R is denoted by lR(S) and defined by lR(S) = {x ∈ R| xS = 0}. Suppose that S = {H(un)un +unG(un) | u ∈ L} and T = {H(xn)xn +xnG(xn) | x ∈ I}, where n ≥ 1 is a fixed integer. In the paper, we investigate the cases when the sets lR(S) and lR(T ) are nonzero.