Generalized derivations and multilinear polynomials in prime rings

Let R be a prime ring with Utumi quotient ring U and extended centroid C, g a nonzero generalized derivation of R, I a nonzero right ideal of R, f (r1,...,rk) a multilinear polynomial over C and n ? 2 be a fixed integer. If g( f (r1,...,rk)n) = g( f (r1,...,rk))n for all r1,...,rk ? I, then one of the following holds: (1) IC = eRC for some idempotent e ? soc(RC) and f (x1,...,xk) is central-valued on eRCe; (2) there exist a,b ? U such that g(x) = ax+xb for all x ? R and (a-?)I = (0), (b-B)I = (0) for some ?,B ? C with (a +b)n-1 = 1; (3) there exists a ? U such that g(x) = ax for all x ? R with aI = (0).