ABSTRACT
Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and let be a multilinear polynomial over C, not central valued on R. Suppose that F and G are non-zero generalized derivations of R and 0≠u0 is an element of R such that
there exists a,c∈U, the right Utumi quotient ring of R, such that F(x) = ax and G(x) = cx, for all x∈R, with u0ac = 0;
there exists a∈U, the right Utumi quotient ring of R, such that F(x) = ax, for all x∈R, with u0a = 0;
there exists a,b,c∈U, the right Utumi quotient ring of R, such that F(x) = ax+xb and G(x) = cx, for all x∈R, with ;
is central valued on R and there exists a,b,c∈U, such that F(x) = ax+xb, G(x) = cx, for all x∈R, with u0(ac+cb) = 0;
there exists a,c∈U and d:R→R a derivation of R such that F(x) = ax+d(x) and G(x) = cx, for all x∈R, with . Moreover, in this case, d is not an inner derivation of R.