• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.619-628
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• 1994

The multiplicity theory initiated by C. Chevalley was the one with respect to ideals generated by a system of parameters of a local ring containing a field [3] and [4]. Samuel generalized the definition to primary ideals belonging the maximal ideal of a local ring which contains a field by a device which used the Hilbert characteristic function [9]. Furthermore Samuel defined multiplicity also in local rings which contain no field [10].

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.407-417
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• 2020

Let (R, m) be a d-dimensional Cohen-Macaulay local ring with infinite residue field. Let I be an ideal of R that has analytic spread ℓ(I) = d, satisfies the G_d condition, the weak Artin-Nagata property AN^-_d-2 and m is not an associated prime of R/I. In this paper, we show that if j₁(I) = λ(I/J) + λ[R/(J_d-1 :_RI+(J_d-2 :_RI+I):_R m^∞)] + 1, then I has almost minimal j-multiplicity, G(I) is Cohen-Macaulay and r_J(I) is at most 2, where J = (x₁, _…, x_d) is a general minimal reduction of I and J_i = (x₁, _…, x_i). In addition, the last theorem is in the spirit of a result of Sally who has studied the depth of associated graded rings and minimal reductions for m-primary ideals.