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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Invariant polynomials of Ore extensions by $ q$-skew derivations


Authors: Chen-Lian Chuang, Tsiu-Kwen Lee and Cheng-Kai Liu
Journal: Proc. Amer. Math. Soc. 140 (2012), 3739-3747
MSC (2010): Primary 16S36, 16N60, 16W25, 16R50
Published electronically: March 12, 2012
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Abstract: Let $ R$ be a prime ring with the symmetric Martindale quotient ring $ Q$. Suppose that $ \delta $ is a quasi-algebraic $ q$-skew $ \sigma $-derivation of $ R$. For a minimal monic semi-invariant polynomial $ \pi (t)$ of $ Q[t;\sigma ,\delta ]$, we show that $ \pi (t)$ is also invariant if $ \textrm {char}\,R=0$ and that either $ \pi (t)-c$ for some $ c\in Q$ or $ \pi (t)^p$ is a minimal monic invariant polynomial if $ \textrm {char}\,R=p\ge 2$. As an application, we prove that any $ R$-disjoint prime ideal of $ R[t;\sigma ,\delta ]$ is the principal ideal $ \langle p(t)\rangle $ for an irreducible monic invariant polynomial $ p(t)$ unless $ \sigma $ or $ \delta $ is X-inner.


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Additional Information

Chen-Lian Chuang
Affiliation: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Email: chuang@math.ntu.edu.tw

Tsiu-Kwen Lee
Affiliation: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Email: tklee@math.ntu.edu.tw

Cheng-Kai Liu
Affiliation: Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan
Email: ckliu@cc.ncue.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11268-7
PII: S 0002-9939(2012)11268-7
Keywords: Prime ring, (semi-)invariant polynomial, $q$-skew $𝜎$-derivation.
Received by editor(s): June 29, 2010
Received by editor(s) in revised form: April 28, 2011
Published electronically: March 12, 2012
Additional Notes: The first two authors are members of the Mathematics Division, NCTS (Taipei Office).
Communicated by: Harm Derksen
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.