Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Reduction numbers for ideals of higher analytic spread

Sam Huckabaa1

a1 Department of Mathematics, University of Kentucky, Lexington, KY 40506, U.S.A.

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 xs2286 I is a minimal reduction of I}.

(Received October 14 1986)

(Revised December 10 1986)