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2018 On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length
Amir Mafi
J. Commut. Algebra 10(4): 547-557 (2018). DOI: 10.1216/JCA-2018-10-4-547

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}.

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Amir Mafi. "On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length." J. Commut. Algebra 10 (4) 547 - 557, 2018. https://doi.org/10.1216/JCA-2018-10-4-547

Information

Published: 2018
First available in Project Euclid: 16 December 2018

zbMATH: 07003226
MathSciNet: MR3892146
Digital Object Identifier: 10.1216/JCA-2018-10-4-547

Subjects:
Primary: 13A30, 13D40, 13H10

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.10 • No. 4 • 2018
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