Generalized skew derivations and generalization of commuting maps on prime rings

Let R be a prime ring of characteristic different from 2, Q(r) be the right Martindale quotient ring of R and C = Z(Q(r)) be the extended centroid of R. Suppose that f (x(1), ... , x(n)) is a noncentralmultilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If F(u)u - G(u)F(u) = 0 for all u is an element of f (R), then one of the following holds: (1) there exist a, p is an element of Q(r) such that F(x) = ax and G(x) = pxp(-1) for all x is an element of R, with p(-1)a is an element of C; (2) there exist a, c, p is an element of Q(r) such that F(x) = ax + pxp(-1)c and G(x) = pxp(-1) for all x is an element of R, with f (R)(2) subset of C and p(-1)(a - c) is an element of C; (3) there exist a, p is an element of Q(r) such that F(x) = ax - pxp(-1)a and G(x) = - pxp(-1) for all x is an element of R, with f (R)(2) subset of C.