Generalized derivations and commuting additive maps on multilinear polynomials in prime rings

Abstract

Let $R$ be a prime ring with characteristic different from $2, U$ be its right Utumi quotient ring, $C$ be its extended centroid, $F$ and $G$ be additive maps on $R$ , $f(x_1, ..., x_n)$ be a multilinear polynomial over $C$, and $I$ be a nonzero right ideal of $R$ . We obtain information about the structure of $R$ and describe the form of $F$ and $G$ in the following cases: $$(1) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ for all $r_1, . . . , r_n \in R$, where $F$ and $G$ are generalized derivations of $R$ ; $$(2) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ for all $r_1, ..., r_n \in I$, where $F$ and $G$ are derivations of $R$