Generalized skew-derivations as generalizations of Jordan homomorphisms in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, Qr be the right Martindale quotient ring of R and (Formula presented.) be the extended centroid of R. Suppose that (Formula presented.) is a noncentral multilinear polynomial over C and (Formula presented.) G are two generalized skew-derivations of R associated to the same automorphism α. If (Formula presented.) for all (Formula presented.) then one of the following holds: There exist (Formula presented.) and (Formula presented.) such that (Formula presented.) and (Formula presented.) for all (Formula presented.) There exists (Formula presented.) such that F(x) = ax for all (Formula presented.) and G = Id, the identity map. There exist (Formula presented.) such that F(x) = ax and G(x) = xb for all (Formula presented.) with (Formula presented.) and (Formula presented.) is central valued on R. © 2022 Taylor & Francis Group, LLC.