Ratliff-Rush closures and linear growth of primary decompositions of ideals

TL;DR

This paper studies the stabilization of associated primes in Ratliff-Rush closures of ideals and proves that these closures have linear growth in their primary decompositions, extending previous results.

Contribution

It extends the understanding of associated prime stabilization and establishes linear growth of primary decompositions for Ratliff-Rush closures with respect to modules.

Findings

Sequences of associated primes are increasing and stabilize.

Ratliff-Rush closures have linear growth in primary decompositions.

Provides a characterization of the set ^*(I,E).

Abstract

Let $R$ be a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\Ass_RE/ \widetilde{I_E^n}$ and $\Ass_R\widetilde{I^n _E}/\widetilde{I^{n+1}_E}$, $n = 1,2, \dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\widetilde{A^*}(I,E)$ is included. A second purpose of this paper is to prove that $I$ has linear growth primary decompositions for Ratliff-Rush closures with respect to $E$, that is, there exists a positive integer $k$ such that for every positive integer $n$, there exists a minimal primary decomposition $\widetilde {I^n_E}= Q_1\cap \cdots\cap Q_s$ in $E$ with $(\Rad(Q_i:_RE))^{nk}\subseteq (Q_i:_R E)$, for all $i= 1, \dots, s$.