LAUSR

Abstract Let R be a prime ring of characteristic different from two with Utumi quotient ring U and extended centroid C, be a multilinear polynomial over C, which is not central valued on R. Suppose that F, G and H are three generalized derivations on R. If for all , then one of the following holds: (i) G = 0 and H = 0, (ii) F = 0 and H = 0, (iii) there exist such that F(x)=ax, and for all , (iv) there exist such that F(x)=xa, G(x)=bx and H(x)=abx for all , (v) there exist such that F(x)=ax, G(x)=xb and H(x)=xab for all , (vi) is central valued on R and one of the following holds: (a) there exist such that F(x)=ax, G(x)=xb and H(x)=xab for all , (b) there exist such that , G(x)=cx and for all .