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The Briançon-Skoda theorem and coefficient ideals for non- $ \mathfrak{m}$-primary ideals


Authors: Ian M. Aberbach and Aline Hosry
Journal: Proc. Amer. Math. Soc. 139 (2011), 3903-3907
MSC (2010): Primary 13A35, 13H05
DOI: https://doi.org/10.1090/S0002-9939-2011-10871-2
Published electronically: March 28, 2011
MathSciNet review: 2823036
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Abstract: We generalize a Briançon-Skoda type theorem first studied by Aberbach and Huneke. With some conditions on a regular local ring $ (R,\mathfrak{m})$ containing a field, and an ideal $ I$ of $ R$ with analytic spread $ \ell$ and a minimal reduction $ J$, we prove that for all $ w \geq -1$, $ \overline{I^{\ell+w}} \subseteq J^{w+1} \mathfrak{a} (I,J),$ where $ \mathfrak{a}(I,J)$ is the coefficient ideal of $ I$ relative to $ J$, i.e. the largest ideal $ \mathfrak{b}$ such that $ I\mathfrak{b}=J\mathfrak{b}$. Previously, this result was known only for $ \mathfrak{m}$-primary ideals.


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Additional Information

Ian M. Aberbach
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: aberbachi@missouri.edu

Aline Hosry
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: aline.hosry@mizzou.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10871-2
Received by editor(s): October 5, 2010
Published electronically: March 28, 2011
Communicated by: Irena Peeva
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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