A note on the quintasymptotic prime ideals

TL;DR

This paper explores the structure of quintasymptotic prime ideals in Noetherian rings, providing a new description using Rees valuation rings and revisiting properties of ideal closures.

Contribution

It offers a novel characterization of the quintasymptotic prime ideals via Rees valuation rings, complementing existing descriptions based on Rees rings.

Findings

New description of $ar{A^*}(I)$ using Rees valuation rings

Reproof of stability of ideal closures under localization

Enhanced understanding of asymptotic prime ideals

Abstract

Let $R$ denote a commutative Noetherian ring, $I$ an ideal of $R$, and let $S$ be a multiplicatively closed subset of $R$. In \cite{Ra1}, Ratliff showed that the sequence of sets ${\rm Ass}_RR/\bar{I}\subseteq {\rm Ass}_RR/\bar{I^2} \subseteq {\rm Ass}_R R/\bar{I^3}\subseteq \dots $ increases and eventually stabilizes to a set denoted $\bar{A^\ast}(I)$. In \cite{Mc2}, S. McAdam gave an interesting description of $\bar{A^\ast}(I)$ by making use of $R[It,t^{-1}]$, the Rees ring of $I$. In this paper, we give a second description of $\bar{A^\ast}(I)$ by making use of the Rees valuation rings of $I$. We also reprove a result concerning when $\bar{I^n}R_S\cap R=\bar{I^n}$ for all integers $n>0$.