On X-generalized skew derivations and their applications

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, alpha be an automorphism of R, d be a skew derivation of R with associated automorphism alpha, F and G be two nonzero X-generalized skew derivation of R with associated term (b, alpha, d) and (b', alpha, d), respectively, S be the set of the evaluations of f (x(1),...,x(n)) on R, where f (x(1),..., x(n)) is a non-central multilinear polynomial over C in n non-commuting variables. Let 0 not equal v is an element of R be such that F(x)x + G(x)xv = 0 for all x is an element of S. Then one of the following statements holds:(a) (x(1),..., x(n))(2) is central valued on R and there exist a, a' is an element of Q(r) such that F(x) = G(x) = a'x for any x is an element of R with a+ a'v = 0;(b) v is an element of C and F = -vG.In the last part of the paper, we present some applications on the basis of the foregoing proposed result.