A note on the quintasymptotic 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 ${\mathrm{Ass}}_{R}R/\bar{I}\subseteq {\mathrm{Ass}}_{R}R/\overline{I^2}\subseteq {\mathrm{Ass}}_{R}R/\overline{I^3}\subseteq \dots$ increases and eventually stabilizes to a set denoted $\overline{A^{\ast }}(I)$. In \cite{Mc2}, S. McAdam gave an interesting description of $\overline{A^{\ast }}(I)$ by making use of $R[It,t^{-1}]$, the Rees ring of $I$. In this paper, we give a second description of $\overline{A^{\ast }}(I)$ by making use of the Rees valuation rings of $I$. We also reprove a result concerning when $\overline{I^n}{R}_S\cap R=\overline{I^n}$ for all integers $n>0$.

arXiv comment5 pages, to appear in Journal of Pure and Applied Algebra