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Ratliff-Rush ideal and reduction numbers. (English) Zbl 1391.13019
Let $$(R,\mathfrak{m})$$ be a Cohen-Macaulay local ring of positive dimension $$d$$ and infinite residue field. Let $$I$$ be an $$\mathfrak{m}$$-primary ideal and $$J$$ a minimal reduction of $$I$$. The author shows that $$\widetilde{r_{J}(I)} \leq r_{J}(I)$$. This answer to a question that made by M. E. Rossi and I. Swanson [Contemp. Math. 331, 313–328 (2003; Zbl 1089.13501), Question 4.6]. The author shows that every minimal reduction of $$I$$ can be generated by a tame supercial sequence of $$I$$ and uses inductive arguments for to show the most result.

MSC:
 13C14 Cohen-Macaulay modules 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Keywords:
reduction number; Ratliff-Rush ideal
Macaulay2
Full Text:
References:
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