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 x_1,...,x_n in I, then one of the following holds: (1) aI=ag(I)=0; (2) g(x)=bx+[c,x] for all xin R, where b,cin U. In this case either [c,I]I=0=abI or aI=0=a(b+c)I; (3) [f(X_1,...,X_n),X_{n+1}]X_{n+2} is an identity for I.