Identities with product of generalized derivations of prime rings

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, f(x1,...,x n) a multilinear polynomial over C which is not an identity for R, F and G two non-zero generalized derivations of R. If F(u)G(u)=0 for all u f(R)= f(r1,...,rn): ri R, then one of the following holds: (i) There exist a, c U such that ac=0 and F(x)=xa, G(x)=cx for all x R; (ii) f(x1,...,xn)2 is central valued on R and there exist a, c U such that ac=0 and F(x)=ax, G(x)=xc for all x R; (iii) f(x1,...,xn) is central valued on R and there exist a,b,c,q U such that (a+b)(c+q)=0 and F(x)=ax+xb, G(x)=cx+xq for all x R. © 2013 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.