On the computation of the Ratliff-Rush closure, associated graded ring\n and invariance of a length

2015/11/02 by Амир Мафи, Mafi, Amir
Mathematics · #13A30 #13D40 #13H10 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1511.00402

openalex publication_date 2015/11/02 · openalex created_date 2022/09/22 · openalex updated_date 2026/07/28

Abstract

Let (R, fm) be a Cohen-Macaulay local ring of positive dimension d and\ninfinite residue field. Let I be an fm-primary ideal of R and J be a\nminimal reduction of I. In this paper we show that if widetildeI^k=I^k\nand J\∩ Iⁿ=JI^n-1 for all n\≥ k+2, then widetildeIⁿ=Iⁿ for\nall n\≥ k. As a consequence, we can deduce that if r_J(I)=2, then\n widetildeI=I if and only if widetildeIⁿ=Iⁿ for all n\≥ 1.\nMoreover, we recover some main results [ refCpv] and [ refG]. Finally, we\ngive a counter example for question 3 of [ refP1].\n

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