On X-generalized skew derivations and their applications

Let R be a prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid, α be an automorphism of R, d be a skew derivation of R with associated automorphism α, F and G be two nonzero X-generalized skew derivation of R with associated term (b, α, d) and (b^́, α, d), respectively, S be the set of the evaluations of f(x1, . . ., xn) on R, where f(x1, . . ., xn) is a non-central multilinear polynomial over C in n non-commuting variables. Let 0 /= v ∈ R be such that F(x)x + G(x)xv = 0 for all x ∈ S. Then one of the following statements holds: (a) f(x1, . . ., xn)² is central valued on R and there exist a, a^́ ∈ Qr such that F(x) = ax, G(x) = a^́x for any x ∈ R with a + a^́v = 0; (b) v ∈ C and F = −vG. In the last part of the paper, we present some applications on the basis of the foregoing proposed result.