Vanishing Derivations on Some Subsets in Prime Rings Involving Generalized Derivations

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and f (x(1), ... , x(n)) a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and f (I) = {f (x(1), ... , x(n))|x(1), ... , x(n) is an element of I}. If f (x(1), ... , x(n)) is not central valued on R and d(F-2 (u)u - G(u(2))) = 0 for all u is an element of f (I), then all possible forms of the maps F, G and d are described in the present paper.