Actions of Generalized Derivations on Multilinear Polynomials in Prime Rings

Let K be a commutative ring with unity, R a non-commutative prime K-algebra with center Z(R), U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, I a non-zero two-sided ideal of R, H and G non-zero generalized derivations of R. Suppose that f(x(1), ... , x(n)) is a non-central multilinear polynomial over K such that H(f(X))f(X) - f(X)G(f(X)) = 0 for all X = (x(1), ... , x(n)) is an element of I(n). Then one of the following holds: (1) There exists a is an element of U such that H(x) = xa and G(x) = ax for all x is an element of R. (2) f (x(1), ... , x(n))(2) is central valued on R and there exist a, b is an element of U such that H(x) - ax+xb and G(x) = bx+xa for all x is an element of R. (3) char(R) = 2 and R satisfies s(4), the standard identity of degree 4.