Annihilators and centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, C its extended centroid, a, b∈ R, f(x1, … , xn) a non-central multilinear polynomial over C with n non-commuting variables and G a non-zero generalized skew derivation of R. Assume a≠ 0 , b∉ C, S= f(r1, … , rn) : r1, … , rn∈ R and a[ b, G(x) x] = 0 , for all x∈ S. Then one of the following holds:(a)there exists c∈ Qr such that ac= abc= 0 and G(x) = cx, for any x∈ R;(b)f(x1,…,xn)2 is central valued on R and there exists c∈ Qr su ch that a[ b, c] = 0 and G(x) = cx, for any x∈ R. © 2017, The Managing Editors.