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 l_R(S) and defined by l_R(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 l_R(S) and l_R(T) are nonzero.