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

 

Gorenstein rings and irreducible parameter ideals


Authors: Thomas Marley, Mark W. Rogers and Hideto Sakurai
Journal: Proc. Amer. Math. Soc. 136 (2008), 49-53
MSC (2000): Primary 13D45; Secondary 13H10
Published electronically: September 27, 2007
MathSciNet review: 2350387
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Abstract: Given a Noetherian local ring $ (R,m)$ it is shown that there exists an integer $ \ell$ such that $ R$ is Gorenstein if and only if some system of parameters contained in $ m^{\ell}$ generates an irreducible ideal. We obtain as a corollary that $ R$ is Gorenstein if and only if every power of the maximal ideal contains an irreducible parameter ideal.


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Additional Information

Thomas Marley
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
Email: tmarley@math.unl.edu

Mark W. Rogers
Affiliation: Department of Mathematics, Missouri State University, Springfield, Missouri 65897
Email: markrogers@missouristate.edu

Hideto Sakurai
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 214-8571, Japan
Email: hsakurai@math.meiji.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08958-7
PII: S 0002-9939(07)08958-7
Keywords: Gorenstein, system of parameters, irreducible ideal
Received by editor(s): August 25, 2006
Received by editor(s) in revised form: September 21, 2006
Published electronically: September 27, 2007
Additional Notes: The second author was supported for eight weeks during the summer of 2006 through the University of Nebraska-Lincoln’s Mentoring through Critical Transition Points grant (DMS-0354281) from the National Science Foundation.
Dedicated: Dedicated to Professor Shiro Goto on the occasion of his sixtieth birthday
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.