Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1,…,xn) be a noncentral multilinear polynomial over C, m≥1 a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag(f(r1,…,rn))m=0 for all r1,…,rn∈I, then one of the following holds:
- (1)aI=ag(I)=(0);
- (2)g(x)=qx, for some q∈U and aqI=0;
- (3)[f(x1,…,xn),xn+1]xn+2 is an identity for I;
- (4)g(x)=cx+[q,x] for all x∈R, where c,q∈U such that cI=0 and [q,I]I=0.