Commuting and Centralizing Generalized Derivations on Lie Ideals in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] k x − x[G(x), x] k = 0 for any x ∈ L, then one of the following holds:1) either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R2) or R satisfies the standard identity s 4(x 1, …, x 4) and one of the following conclusions occurs(a) there exist a, b, c, q ∈ U, such that a −b + c −q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R(b) there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc−d(x) for all x ∈ R, with a + b − c ∈ C.