Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Abstract

Let R be a prime ring of characteristic different from 2 and 3, Q _r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.

We prove that, if F is a nonzero generalized skew derivation of R such that F(x)×[F(x), x]ⁿ = 0 for any x ∈ L, then either there exists λ ∈ C such that F(x) = λx for all x ∈ R, or R ⊆ M ₂(C) and there exist a ∈ Q _r and λ ∈ C such that F(x) = ax + xa + λx for any x ∈ R.