Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2012-11268-7
Published electronically: March 12, 2012
MathSciNet review: 2944714
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, ``Rings with Generalized Identities'', Marcel Dekker, Inc., New York-Basel-Hong Kong, 1996. MR 1368853 (97g:16035)
  • 2. A. D. Bell, When are all prime ideals in an Ore extension Goldie?, Comm. Algebra 13(8) (1985), 1743-1762. MR 792560 (86j:16003)
  • 3. W. Chin, Prime ideals in differential operator rings and crossed products of infinite groups, J. Algebra 106 (1987), 78-104. MR 878469 (88c:16039)
  • 4. C.-L. Chuang and T.-K. Lee, Algebraic $ q$-skew derivations, J. Algebra 282 (2004), 1-22. MR 2095569 (2005g:16068)
  • 5. C.-L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), 59-77. MR 2138371 (2006c:16059)
  • 6. C.-L. Chuang and T.-K. Lee, Derivations and skew polynomial rings, Comm. Algebra 35 (2007), 527-539. MR 2294614 (2007k:16063)
  • 7. C.-L. Chuang and T.-K. Lee, Ore extensions which are GPI-rings, Manuscripta Math. 124 (2007), 45-58. MR 2336054 (2009a:16048)
  • 8. C.-L. Chuang and Y.-T. Tsai, On the structure of semi-invariant polynomials in Ore extensions, J. Algebra 322(7) (2009), 2464-2491. MR 2553690 (2010m:16039)
  • 9. C.-L. Chuang and Y.-T. Tsai, Higher derivations of Ore extensions by $ q$-skew derivations,
    J. Pure and Applied Algebra 214(10) (2010), 1778-1786. MR 2608105 (2011e:16050)
  • 10. E. Cisneros, M. Ferrero and M. I. Gonzalez, Prime ideals of skew polynomial rings and skew Laurent polynomial rings, Math. J. Okayama Univ. 32 (1990), 61-72. MR 1112011 (92g:16041)
  • 11. M. Ferrero and J. Matczuk, Prime ideals in skew polynomial rings of derivation type, Comm. Algebra 18(3) (1990), 689-710. MR 1052761 (91h:16007)
  • 12. M. Ferrero, Prime and principal closed ideals of polynomial rings, J. Algebra 134(1) (1990), 45-59. MR 1068414 (91h:16008)
  • 13. N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996. MR 1439248 (98a:16024)
  • 14. V. K. Kharchenko, Generalized identities with automorphisms, Algebra i Logika 14 (1975), 132-148. MR 0399153 (53:3004)
  • 15. V. K. Kharchenko and A. Z. Popov, Skew derivations of prime rings, Comm. Algebra 20(11) (1992), 3321-3345. MR 1186710 (93k:16071)
  • 16. T. Y. Lam and A. Leroy, Algebraic conjugacy classes and skew polynomial rings, ``Proceedings of the NATO Advanced Research Workshop on Perspectives in Ring Theory, Antwerp, Belgium, July 19-29, 1987'', Kluwer, Dordrecht, 1988. MR 1048406 (91c:16015)
  • 17. T. Y. Lam, A. Leroy, K. H. Leung and J. Matczuk, Invariant and semi-invariant polynomials in skew polynomial rings, Israel Mathematics Conference Proceedings 1 (1989), 247-261. MR 1029317 (90k:16004)
  • 18. T. Y. Lam and A. Leroy, Homomorphisms between Ore extensions, Contemporary Math. 124, Amer. Math. Soc., Providence, RI, 1992, 83-110. MR 1144030 (93b:16052)
  • 19. A. Leroy and J. Matczuk, Prime ideals of Ore extensions, Comm. Algebra 19(7) (1991), 1893-1907. MR 1121112 (92h:16029)
  • 20. A. Leroy and J. Matczuk, The extended centroid and X-inner automorphisms of Ore extensions, J. Algebra 145 (1992), 143-177. MR 1144664 (93b:16053)
  • 21. J. Matczuk, Extended centroid of skew polynomial rings, Math. J. Okayama Univ. 30 (1988), 13-20. MR 976726 (89m:16006)
  • 22. D. S. Passman, ``Infinite Crossed Product'', Pure and Applied Mathematics, 135. Academic Press, Inc., Boston, MA, 1989. MR 979094 (90g:16002)
  • 23. D. S. Passman, Prime ideals in enveloping rings, Trans. Amer. Math. Soc. 302(2) (1987), 535-560. MR 891634 (88f:17015)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16S36, 16N60, 16W25, 16R50

Retrieve articles in all journals with MSC (2010): 16S36, 16N60, 16W25, 16R50


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: https://doi.org/10.1090/S0002-9939-2012-11268-7
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.

American Mathematical Society