## Taiwanese Journal of Mathematics

### CENTRALIZING GENERALIZED DERIVATIONS ON POLYNOMIALS IN PRIME RINGS

Vincenzo De Filippis

#### 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 $(a-b+\alpha)I = 0$.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 5 (2012), 1847-1863.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406801

Digital Object Identifier
doi:10.11650/twjm/1500406801

Mathematical Reviews number (MathSciNet)
MR2970689

Zentralblatt MATH identifier
1272.16037

#### Citation

Filippis, Vincenzo De. CENTRALIZING GENERALIZED DERIVATIONS ON POLYNOMIALS IN PRIME RINGS. Taiwanese J. Math. 16 (2012), no. 5, 1847--1863. doi:10.11650/twjm/1500406801. https://projecteuclid.org/euclid.twjm/1500406801

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