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Reduction numbers for ideals of higher analytic spread

Published online by Cambridge University Press:  24 October 2008

Sam Huckaba
Sam Huckaba
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, U.S.A.
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Extract

Let (R, M) be a commutative Noetherian local ring having an identity, and assume the residue field R/M is infinite. If I is an ideal in R, recall that an ideal J contained in I is called a reduction of I if JIn = In + 1 for some non-negative integer n. A reduction of J of I is called a minimal reduction of I if it does not properly contain another reduction of I. Reductions (and minimal reductions) were introduced and studied by Northcott and Rees[8]. If J is a reduction of I we define the reduction number of I with respect to J, denoted rJ(I), to be the smallest non-negative integer n such that JIn = In + 1 (note that rJ(I) = 0 if and only if J = I). The reduction number of I (sometimes referred to as the reduction exponent) is defined as r(I) = min{rj(I)|J ⊆ I is a minimal reduction of I}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 1 , July 1987 , pp. 49 - 57
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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