Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Let R be a prime ring with center Z(R) and G be a generalized alpha -derivation of R for alpha is an element of Aut (R). Let a E R be a nonzero element and n be a fixed positive integer. (i) If aG (x)(n) is an element of Z ( R ) for all x is an element of R then aG (x) = 0 for all x is an element of R unless dim(C) RC = 4. (ii) If aG (x)(n) E Z ( R ) for all x is an element of L , where L is a noncommutative Lie ideal of R then aG (x) = 0 for all x is an element of R unless dim(C) RC = 4.