Vanishing Derivations and Centralizers of Generalized Derivations On Multilinear Polynomials

Let R be a prime algebra over a commutative ring K with unity, and let f(x(1),..., x(n)) be a multilinear polynomial over K, not central valued on R. Suppose that d is a nonzero derivation of R and G is a nonzero generalized derivation of R such that d[G(f(r(1),..., r(n))), f(r(1),..., r(n))] = 0 for all r(1),..., r(n) is an element of R. If the characteristic of R is different from 2, then one of the following holds: 1. There exists lambda is an element of C, the extended centroid of R, such that G(x) = lambda x, for all x is an element of R; 2. There exist a is an element of U, the Utumi quotient ring of R, and lambda is an element of C = Z (U) such that G(x) = ax + xa + lambda x, for all x is an element of R, and f(x(1),..., x(n))(2) is central valued on R