Generalized derivations with power-central values on multilinear polynomials

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X-1,..., X-t) a multilinear polynomial over K. If g(f(x(1),...,x(t)))(n) is an element of Z for all x(1),...,x(t) is an element of R, then either there exists an element lambda is an element of C such that g (x) = lambda x for all x is an element of R or f (x(1),..., x(t)) is central-valued on R except when R satisfies s(4), the standard identity in four variables.