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

 

Core of ideals of Noetherian local rings


Author: Hsin-Ju Wang
Journal: Proc. Amer. Math. Soc. 136 (2008), 801-807
MSC (2000): Primary 13H10, 13A15; Secondary 13A30
Published electronically: November 23, 2007
MathSciNet review: 2361851
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Abstract: The core of an ideal is the intersection of all its reductions. In 2005, Polini and Ulrich explicitly described the core as a colon ideal of a power of a single reduction and a power of $ I$ for a broader class of ideals, where $ I$ is an ideal in a local Cohen-Macaulay ring. In this paper, we show that if $ I$ is an ideal of analytic spread $ 1$ in a Noetherian local ring with infinite residue field, then with some mild conditions on $ I$, we have $ \core (I)\supseteq J(J^n: I^n)=I(J^n: I^n)=(J^{n+1}: I^n)\cap I$ for any minimal reduction $ J$ of $ I$ and for $ n\gg 0$.


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Additional Information

Hsin-Ju Wang
Affiliation: Department of Mathematics, National Chung Cheng University, Chiayi 621, Taiwan

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09038-7
PII: S 0002-9939(07)09038-7
Keywords: Core, analytic spread, minimal reduction
Received by editor(s): November 2, 2004
Received by editor(s) in revised form: November 27, 2006
Published electronically: November 23, 2007
Communicated by: Bernd Ulruch
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.