Centralizers of X-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings

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, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial f(x₁, … , x_n) over C with n non-commuting variables. Let u, v∈ R be such that uG(x) x+ G(x) xv= 0 for all x∈ S. Then one of the following statements holds:(a)v∈ C and there exist a, b, c∈ Q_r such that G(x) = ax+ bxc for any x∈ R with (u+ v) a= (u+ v) b= 0 ;(b)f(x1,…,xn)2 is central-valued on R and there exists a∈ Q_r such that G(x) = ax for all x∈ R with ua+ av= 0.