Centralizers of generalized skew derivations on multilinear polynomials

Abstract

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,..., x_n) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r₁,..., r_n): r_i ∈ R} be the set of all evaluations of f(x₁,..., x_n) in R, while A = {[G (f(r₁,..., r_n)), f(r₁,..., 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).