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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a proper nonnilpotent ideal in a local (Noetherian) ring and let be a reduction of ; that is, and for some nonnegative integer . We prove that there exists a finite free local unramified extension ring of such that the ideal has a minimal reduction with the property that the number of elements in a minimal generating set of is equal to the analytic spread of and thus also equal to the analytic spread of .

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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