Centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x 1,..., x n ) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r 1,..., r n ): r i ∈ R} be the set of all evaluations of f(x 1,..., x n ) in R, while A = {[G (f(r 1,..., r n )), f(r 1,..., r n )]: r i ∈ R}, and let C R (A) be the centralizer of A in R; i.e., C R (A) = {a ∈ R: [a, x] = 0, ∀ x ∈ A }. We prove that if A ≠ (0), then C R (A) = Z(R).