Generalized Skew Derivations and Nilpotent Values on Lie Ideals

Let R be a prime ring of characteristic different from 2 and 3, Qr be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x-xG(x))n = 0 for all x L, then one of the following holds: (a) there exists c Qr such that F(x) = xc and G(x) = cx; (b) R satisfies s4 and there exist a, b, c Qr such that F(x) = ax + xc, G(x) = cx + xb and (a-b)2 = 0. © 2019 Academy of Mathematics and Systems Science, Chinese Academy of Sciences.