Generalized skew derivations with centralizer conditions on multilinear polynomials

Let R be a noncommutative prime ring of characteristic not 2 with extended centroid C, the maximal right ring of quotients Q and a nonzero generalized skew derivation d. Assume that f (X-1, center dot center dot center dot ,X-n) is amultilinear polynomial over C that is not central-valued on R and f (R) is the set of all evaluations of the multilinear polynomial f (X-1, center dot center dot center dot ,X-n) in R. Denote the set S := {delta(u)u | u is an element of f (R)}. The goal of the paper is to study C-R(S), the centralizer of S in R. To be precise, given a noncentral element b is an element of R it is proved that if b is an element of C-R(S), i.e.,