Annihilators and centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, Q(r) its right Martindale quotient ring, C its extended centroid, a, b. R, f (x1,..., xn) a non-central multilinear polynomial over C with n non-commuting variables and G a non-zero generalized skew derivation of R. Assume a = 0, b /. C, S = {f (r1,..., rn) : r1,..., rn. R} and a[b, G(x) x] = 0, for all x. S. Then one of the following holds: (a) there exists c. Q(r) such that ac = abc = 0 and G(x) = cx, for any x. R; (b) f (x1,..., xn) 2 is central valued on R and there exists c. Q(r) su ch that a[b, c] = 0 and G(x) = cx, for any x is an element of R.