CENTRALIZING GENERALIZED DERIVATIONS ON POLYNOMIALS IN PRIME RINGS

Abstract

Let $R$ be a prime ring, $Z(R)$ its center, $U$ its right Utumi quotient ring, $C$ its extended centroid, $G$ a non-zero generalized derivation of $R$, $f(x_1,\ldots,x_n)$ a non-zero polynomial over $C$ and $I$ a non-zero right ideal of $R$. If $f(x_1,\ldots,x_n)$ is not central valued on $R$ and $[G(f(r_1,\ldots,r_n)), f(r_1,\ldots,r_n)] \in C$, for all $r_1,\ldots,r_n \in I$, then either there exist $a \in U$, $\alpha \in C$ such that $G(x) = ax$ for all $x \in R$, with $(a-\alpha)I = 0$ or there exists an idempotent element $e \in soc(RC)$ such that $IC = eRC$ and one of the following holds:

1. $f(x_1,\ldots,x_n)$ is central valued in $eRCe$;

2. $char(R) = 2$ and $eRCe$ satisfies the standard identity $s_4$;

3. $char(R) = 2$ and $f(x_1,\dots,x_n)^2$ is central valued in $eRCe$;

4. $f(x_1,\ldots,x_n)^2$ is central valued in $eRCe$ and there exist $a, b \in U$, $\alpha \in C$ such that $G(x) = ax+xb$, for all $x \in R$, with $(a-b+\alpha)I = 0$.