b-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, F≠ 0 an b-generalized skew derivation of R, L a non-central Lie ideal of R, 0 ≠ a∈ R and n≥ 1 a fixed integer. In this paper, we prove the following two results:1.If R has characteristic different from 2 and 3 and a[ F(x) , x] ⁿ= 0 , for all x∈ L, then either there exists an element λ∈ C, such that F(x) = λx, for all x∈ R or R satisfies s₄(x₁, … , x₄) , the standard identity of degree 4, and there exist λ∈ C and b∈ Q, such that F(x) = bx+ xb+ λx, for all x∈ R.2.If char (R) = 0 or char (R) > n and a[ F(x) , x] ⁿ∈ Z(R) , for all x∈ R, then either there exists an element λ∈ C, such that F(x) = λx, for all x∈ R or R satisfies s₄(x₁, … , x₄).