Centralizers of X-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings

Let R be a prime ring of characteristic different from 2, Q(r) be its right Martindale quotient ring and C be its extended centroid, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial f (x1,..., xn) over C with n non-commuting variables. Let u, v is an element of R be such that uG(x) x + G(x) xv = 0 for all x is an element of S. Then one of the following statements holds: (a) v is an element of C and there exist a, b, c. Qr such that G(x) = ax + bxc for any x is an element of R with (u + v) a = (u + v) b = 0; (b) f (x1,..., x(n))(2) is central-valued on R and there exists a is an element of Q(r) such that G(x) = ax for all x is an element of R with ua + av = 0.