Generalized Derivations Vanishing on Co-commutator Identities in Prime Rings

Abstract

Let $R$ be a noncommutative prime ring of char $(R)\neq 2$
with Utumi quotient ring $U$ and extended centroid $C$ and $I$ a nonzero two sided ideal of $R$.
Suppose that $F (\neq 0)$, $G$ and $H$ are three generalized derivations of $R$ and $f(x_1,\ldots,x_n)$ is a multilinear polynomial over $C$,
which is not central valued on $R$. If $$F(G(f(r))f(r)-f(r)H(f(r)))=0$$
for all $r=(r_1,\ldots,r_n) \in I^n$, then we obtain information about the structure of $R$ and describe the all possible forms of the maps $F$, $G$ and $H$.