The strong Rees property of powers of the maximal ideal and Takahashi-Dao's question

TL;DR

This paper introduces the strong Rees property (SRP) for m-primary ideals in Noetherian local rings, proving powers of the maximal ideal have SRP under certain conditions and exploring characterizations of p_g-ideals.

Contribution

It defines the SRP for m-primary ideals, proves powers of the maximal ideal have SRP when the associated graded ring has depth at least 2, and characterizes p_g-ideals in certain local domains.

Findings

Powers of the maximal ideal have SRP if depth of associated graded ring ≥ 2.

Characterization of p_g-ideals in two-dimensional excellent normal local domains.

Proposed conjecture on which m-primary ideals have SRP.

Abstract

In this paper, we introduce the notion of the strong Rees property (SRP) for $\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal $\mathfrak{m}$ has its property if the associated graded ring $G$ of $\mathfrak{m}$ satisfies $\text{depth} \ G \ge 2$. As its application, we characterize two-dimensional excellent normal local domains so that $\mathfrak{m}$ is a $p_g$-ideal. Finally we ask what $\mathfrak{m}$-primary ideals have SRP and state a conjecture which characterizes the case when $\mathfrak{m}^n$ are the only ideals which have SRP.