On the Ratliff-Rush closure of an ideal of a one-dimensional ring

TL;DR

This paper investigates the asymptotic behavior of the Ratliff-Rush closure of ideals in one-dimensional Noetherian rings, focusing on the asymptotic Ratliff-Rush number and its extremal values relative to the reduction number.

Contribution

It characterizes conditions under which the asymptotic Ratliff-Rush number attains extremal values, linking it to the reduction number in one-dimensional rings.

Findings

Bounds for the asymptotic Ratliff-Rush number in terms of the reduction number

Conditions for the extremal values of the asymptotic Ratliff-Rush number

Relationship between the Ratliff-Rush closure and reduction number

Abstract

Let $I$ be an ideal in a Noetherian ring $R$ and let $\widetilde{I}$ be its Ratliff-Rush closure. In this paper we study the asymptotic Ratliff-Rush number, i.e. $h(I)=\min\{n\in\mathbb N_+ \mid I^m=\widetilde{I^m}, \ \forall \ m\ge n\}$, in the one-dimensional case. Since $1\le h(I)\le r(I)$, where $r(I)$ is the reduction number of $I$, we look for conditions that determine the extremal values of $h(I)$.