Annihilators of Skew Derivations with Engel Conditions on Lie Ideals

Abstract

Let R be a prime ring, L a noncentral Lie ideal of R, and a ∈ R. Set [x, y]1 = [x, y] = xy − yx for x, y ∈ R and inductively [x, y]k = [[x, y]k−1, y] for k > 1. Suppose that δ is a nonzero σ-derivation of R such that a[δ(x), x]k = 0 for all x ∈ L, where σ is an automorphism of R and k is a fixed positive integer. Then a = 0 except when char R = 2 and R ⊆ M2(F), the 2 × 2 matrix ring over a field F.