Reductions of ideals in local rings with finite residue fields
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- by William J. Heinzer, Louis J. Ratliff, Jr. and David E. Rush PDF
- Proc. Amer. Math. Soc. 138 (2010), 1569-1574 Request permission
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 $I$.References
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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
- 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
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2587440