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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
DOI: https://doi.org/10.1090/S0002-9939-07-08958-7
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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  • [BH] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics no. 39, Cambridge, Cambridge University Press, 1993. MR 1251956 (95h:13020)
  • [G] Goto, S., Approximately Cohen-Macaulay rings, J. Algebra 76 (1982), 214-225. MR 659220 (84h:13033)
  • [GSa1] Goto, S. and Sakurai, H., The equality $ I^2 = QI$ in Buchsbaum rings, Rend. Sem. Mat. Univ. Padova 110 (2003), 25-56. MR 2033000 (2004m:13061)
  • [GSa2] Goto, S. and Sakurai, H., Index of Reducibility of Parameter Ideals for Modules Possessing Finite Local Cohomology Modules, Preprint.
  • [Gr] Grothendieck, A., Local Cohomology, notes by R. Hartshorne, Lect. Notes Math. no. 41, Springer, Berlin, 1966. MR 0224620 (37:219)
  • [Ho] Hochster, M., Cyclic purity versus purity in excellent Noetherian rings, Trans. Am. Math. Soc. 231 (1977), no. 2, 463-488. MR 0463152 (57:3111)
  • [Hu] Huneke, C., Tight closure, parameter ideals, and geometry, in Six Lectures on Commutative Algebra (J. Elias, J.M. Giral, R.M. Miró-Roig, and S. Zarzuela, eds.), Progress in Mathematics, vol. 166, Birkhäuser Verlag, Basel, 1998, 187-239. MR 1648666 (99j:13001)
  • [LR] Liu, J. C. and Rogers, M., The index of reducibility of parameter ideals and mostly zero finite local cohomologies, Comm. Algebra, to appear.
  • [Mat] Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics no. 8, Cambridge, Cambridge University Press, 1986. MR 879273 (88h:13001)
  • [NR] Northcott, D. G. and Rees, D., Principal Systems, Quart. J. Math. 8 (1957), 119-127. MR 0096649 (20:3132)
  • [R] Rogers, M., The index of reducibility for parameter ideals in low dimension, J. Alg., 278/2 (2004), 571-584. MR 2071653 (2005e:13026)
  • [St] Strooker, J. R., Homological questions in local algebra, London Math. Soc. Lecture Note series 145, Cambridge Univ. Press, Cambridge 1990. MR 1074178 (91m:13013)

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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: https://doi.org/10.1090/S0002-9939-07-08958-7
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.

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