Generalized derivations of prime rings on multilinear polynomials with annihilator conditions

Let K be a commutative ring with unity, R be a prime K-algebra with characteristic not 2, U be the right Utumi quotient ring of R, C the extended centroid of R, I a nonzero right ideal of R and a a fixed element of R. Let g be a generalized derivation of R and f(X-1,..., X-n) a multilinear polynomial over K. If ag(f(x(1),...,x(n)))f(x(1),...,x(n)) = 0 for all x1,..., x(n) is an element of I, then one of the following holds: (1) aI = ag(I) --= 0; (2) g(x) = bx [c,x] for all x is an element of R, where b,c is an element of U. In this case either [c,I]I = 0 = abI or aI = 0 = a(b + c)I; (3) [f,(X-1,...,X-n),Xn+1]Xn+2 is an identity for I.