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

TL;DR

This paper investigates the conditions under which the Ratliff-Rush closure of powers of an ideal stabilizes, providing new criteria and counterexamples in the context of Cohen-Macaulay local rings.

Contribution

It establishes new conditions for the invariance of Ratliff-Rush closures of ideal powers and relates these to minimal reductions and reduction numbers.

Findings

If certain intersection conditions hold, then the Ratliff-Rush closure stabilizes from a specific power onward.

For reduction number 2, the Ratliff-Rush closure of the ideal equals the ideal if and only if all higher powers' closures equal the powers.

Provides counterexamples to previously posed questions about the invariance of Ratliff-Rush closures.

Abstract

Let $(R,\fm)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\fm$-primary ideal of $R$ and $J$ 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 [\ref{Cpv}] and [\ref{G}]. Finally, we give a counter example for question 3 of [\ref{P1}].