Power Values of Generalized Skew Derivations with Annihilator Conditions on Lie Ideals

Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations d and of R in case of p(d(u)l1 (u)l2d(u)l3 (u)l4 center dot center dot center dot (u)lk)n = 0, for all u. L, where 0 = p. R, l1, l2,..., lk are fixed nonnegative integers with l1 = 0 and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if pd(u)n = 0 for all u. L, then there exists a. Q such that d( x) = ax for all x. R and pa = 0 unless R satisfies s(4).