Journal of Commutative Algebra

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

Amir Mafi

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Abstract

Let $(R,\frak{m} )$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\frak{m} $-primary ideal of $R$, and let $J$ be a minimal reduction of $I$. In this paper, we show that, if $\widetilde {I^k}=I^k$ and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde {I^n}=I^n$ for all $n\geq k$. As a consequence, we can deduce that, if $r_J(I)=2$, then $\widetilde {I}=I$ if and only if $\widetilde {I^n}=I^n$ for all $n\geq 1$. Moreover, we recover some main results of \cite {Cpv, G}. Finally, we give a counter example for Puthenpurakal [Question 3]{P1}.

Article information

Source
J. Commut. Algebra, Volume 10, Number 4 (2018), 547-557.

Dates
First available in Project Euclid: 16 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1544950831

Digital Object Identifier
doi:10.1216/JCA-2018-10-4-547

Mathematical Reviews number (MathSciNet)
MR3892146

Zentralblatt MATH identifier
07003226

Subjects
Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Keywords
Ratliff-Rush filtration Minimal reduction Associated graded ring

Citation

Mafi, Amir. On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length. J. Commut. Algebra 10 (2018), no. 4, 547--557. doi:10.1216/JCA-2018-10-4-547. https://projecteuclid.org/euclid.jca/1544950831


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