An Engel condition with generalized derivations on multilinear polynomials

Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x (1),..., x (n)) a multilinear polynomial over C, I a nonzero right ideal of R. If [g(f(r (1),..., r (n))), f(r (1),..., r (n))] = 0, for all r (1), ..., r (n) epsilon I, then either g(x) = ax, with (a - gamma)I = 0 and a suitable gamma epsilon C or there exists an idempotent element e epsilon soc(RC) such that IC = eRC and one of the following holds: (i) f(x (1),..., x (n)) is central valued in eRCe; (ii) g(x) = cx + xb, where (c+b+alpha)e = 0, for alpha epsilon C, and f (x (1),..., x (n))(2) is central valued in eRCe; (iii) char(R) = 2 and s (4)(x (1), x (2), x (3), x (4)) is an identity for eRCe.