An engel condition with generalized derivations on Lie ideals

Abstract

Let R be a prime ring, with extended centroid C, g a non-zero generalized derivation of R, L a non-central Lie ideal of R, k ≥ 1 a fixed integer. If [g(u), u]k = 0, for all u, then either g(x) = ax, with a ∈ C or R satisfies the standard identity s4. Moreover in the latter case either char(R) = 2 or char(R) = 2 and g(x) = ax + xb, with a, b ∈ Q and a-b∈C. We also prove a more generalized version by replacing L with the set [I, I], where I is a right ideal of R.