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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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