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

 

Factorization of an integrally closed ideal
in two-dimensional regular local rings


Author: Mee-Kyoung Kim
Journal: Proc. Amer. Math. Soc. 125 (1997), 3509-3513
MSC (1991): Primary 13A18; Secondary 13B20, 13C05
MathSciNet review: 1415330
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Abstract: Let $(R,m,k)$ be a two-dimensional regular local ring with algebraically closed residue field $k$ and $I$ be an $m$-primary integrally closed ideal in $R$. Let $T(I)$ be the set of Rees valuations of $I$ and $k(v)$ be the residue field of the valuation ring $V$ associated with $v\in T(I)$. Assume that $(a,b)$ is any minimal reduction of $I$. We show that if $I$ is the product of the distinct simple $m$-primary integrally closed ideals in $(R,m,k)$, then $k(v)$ is generated by the image of $a/b$ over $k$ for all $v\in T(I)$, and the converse of this is also true.


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Additional Information

Mee-Kyoung Kim
Email: mkkim@yurim.skku.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04064-1
PII: S 0002-9939(97)04064-1
Received by editor(s): July 16, 1993
Received by editor(s) in revised form: July 12, 1996
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1997 American Mathematical Society