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)

 

 

Product of distinct simple integrally closed ideals in 2-dimensional regular local rings


Author: Mee-Kyoung Kim
Journal: Proc. Amer. Math. Soc. 125 (1997), 315-321
MSC (1991): Primary 13H05
DOI: https://doi.org/10.1090/S0002-9939-97-03886-0
MathSciNet review: 1396984
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $(R,m)$ be a two-dimensional regular local ring and $I$ an $m$-primary integrally closed ideal in $R$. In this paper, we give equivalent conditions for $I$ to be a product of distinct simple $m$-primary integrally closed ideals (i.e., $I = I_{1}\cdots I_{l}$, where $I_{1},\cdots ,I_{l}$ are distinct simple $m$-primary integrally closed ideals of $R$) in terms of the regularity of $R[It]/p$ for all $p \in \text {Min} (mR[It])$ and in terms of how to choose a minimal generating set for $I$ over its minimal reductions.


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

  • 1. C. Huneke, Complete ideals in two-dimensional regular local rings, Microprogram in Commutative Algebra, MSRI, Springer-Verlag (1987).
  • 2. Craig Huneke and Judith D. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), no. 2, 481–500. MR 943272, https://doi.org/10.1016/0021-8693(88)90274-8
  • 3. Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 0276239
  • 4. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • 5. Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
  • 6. D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Phil. Soc. 50 (1954), 145-158. MR 15:596a
  • 7. Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR 0120249

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13H05

Retrieve articles in all journals with MSC (1991): 13H05


Additional Information

Mee-Kyoung Kim
Affiliation: Department of Mathematics, Sung Kyun Kwan University, Suwon 440-746, Korea
Email: mkkim@yurim.skku.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-97-03886-0
Received by editor(s): April 28, 1993
Communicated by: Eric M. Friedlander
Article copyright: © Copyright 1997 American Mathematical Society