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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 102, Issue 1
  • July 1987, pp. 49-57

Reduction numbers for ideals of higher analytic spread

  • Sam Huckaba (a1)
  • DOI:
  • Published online: 24 October 2008

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)|JI is a minimal reduction of I}.

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[1]P. Eakin and A. Sathaye . Prestable ideals. J. Algebra 41 (1976), 439454.

[4]C. Huneke . On the symmetric and Rees algebra of an ideal generated by a d-sequence. J. Algebra 62 (1980), 268275.

[5]J. Lipman . Stable ideals and Arf rings. Amer. J. Math. 97 (1975), 791813.

[7]A. Micali . Sur les algèbres universelles. Ann. Inst. Fourier, 14 (1964), 3388.

[11]J. D. Sally . Cohen-Macaulay local rings of embedding dimension e + d − 2. J. Algebra 83 (1983), 393408.

[12]J. D. Sally and W. Vasconcelos . Stable rings. J. Pure Appl. Algebra 4 (1974), 319336.

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