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(x1,.., xn) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = f(r1,.., rn): ri ∈ R be the set of all evaluations of f(x1,.., xn) in R, while A = [G (f(r1,.., rn)), f(r1,.., rn)]: ri ∈ R, and let CR(A) be the centralizer of A in R; i.e., CR(A) = a ∈ R: [a, x] = 0, ∀x ∈ A . We prove that if A ≠ (0), then CR(A) = Z(R). © 2017, Pleiades Publishing, Ltd.