Taiwanese Journal of Mathematics

CENTRALIZING GENERALIZED DERIVATIONS ON POLYNOMIALS IN PRIME RINGS

Vincenzo De Filippis

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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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406801

Digital Object Identifier
doi:10.11650/twjm/1500406801

Mathematical Reviews number (MathSciNet)
MR2970689

Zentralblatt MATH identifier
1272.16037

Subjects
Primary: 16N60: Prime and semiprime rings [See also 16D60, 16U10] 16W25: Derivations, actions of Lie algebras

Keywords
prime rings differential identities generalized derivations

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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