Generalized Skew Derivations with Nilpotent Values in Prime Rings

Abstract

Let ℛ be a prime ring, 𝒞 the extended centroid of ℛ, ℒ a Lie ideal of ℛ, F be a nonzero generalized skew derivation of ℛ with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) − yx) n = 0 for all x, y ∈ ℒ, then ℒ is commutative and one of the following statements holds: (1) Either ℒ is central; (2) Or ℛ ⊆ M 2(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.