Generalized Skew Derivations As Jordan Homomorphisms On Multilinear Polynomials

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. Suppose that G is a nonzero generalized skew derivation of R, alpha is the associated automorphism of G, f(x(1),...,x(n)) is a non-central multilinear polynomial over C with n non-commuting variables and S = {f(r(1),...,r(n)) vertical bar r(1),...,r(n) is an element of R.}. If G acts as a Jordan homomorphism on S, then either G(x) = x for all x is an element of R., or G = alpha.