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Reductions of ideals in local rings with finite residue fields


Authors: William J. Heinzer, Louis J. Ratliff Jr. and David E. Rush
Journal: Proc. Amer. Math. Soc. 138 (2010), 1569-1574
MSC (2000): Primary 13A15, 13E05, 13H10
DOI: https://doi.org/10.1090/S0002-9939-10-10050-1
Published electronically: January 8, 2010
MathSciNet review: 2587440
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Abstract: Let $ I$ be a proper nonnilpotent ideal in a local (Noetherian) ring $ (R,M)$ and let $ J$ be a reduction of $ I$; that is, $ J$ $ \subseteq$ $ I$ and $ JI^n$ $ =$ $ I^{n+1}$ for some nonnegative integer $ n$. We prove that there exists a finite free local unramified extension ring $ S$ of $ R$ such that the ideal $ IS$ has a minimal reduction $ K$ $ \subseteq$ $ JS$ with the property that the number of elements in a minimal generating set of $ K$ is equal to the analytic spread of $ K$ and thus also equal to the analytic spread of $ I$.


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Additional Information

William J. Heinzer
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
Email: heinzer@math.purdue.edu

Louis J. Ratliff Jr.
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Email: ratliff@math.ucr.edu

David E. Rush
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Email: rush@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10050-1
Keywords: Minimal reduction, analytic spread, finite free local unramified extension, Rees valuation rings, projective equivalence of ideals
Received by editor(s): February 13, 2009
Received by editor(s) in revised form: May 26, 2009
Published electronically: January 8, 2010
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 American Mathematical Society

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