On Identities with Composition of Generalized Derivations

Let R be a prime ring with extended centroid C, Q maximal right ring of quotients of R, RC central closure of R such that dim(C) (RC) > 4, f(X-1,...,X-n) a multilinear polynomial over C that is not central-valued on R, and f (R) the set of all evaluations of the multilinear polynomial f(X-1,...,X-n) in R. Suppose that G is a nonzero generalized derivation of R such that G(2) (u) u is an element of C for all u is an element of f (R). Then one of the following conditions holds: