# On the index of reducibility of parameter ideals: The stable and limit values

## TL;DR

This paper extends Northcott's theorem by defining and analyzing the stable and limit values of the index of reducibility of parameter ideals for any finitely generated module over a Noetherian local ring, beyond Cohen-Macaulay modules.

## Contribution

It introduces the concepts of stable and limit values of the index of reducibility for parameter ideals, generalizing Northcott's invariant to all finitely generated modules.

## Key findings

- Defines the stable value of the index of reducibility.
- Introduces the limit value of the index of reducibility.
- Extends Northcott's theorem to non-Cohen-Macaulay modules.

## Abstract

Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. A famous result of Northcott says that if $M$ is Cohen-Macaulay, then the index of reducibility of parameter ideals on $M$ is an invariant of the module. The aim of this paper is to extend Northcott's theorem for any finitely generated $R$-module. We call this invariant the stable value of the indices of reducibility of parameter ideals of $M$. We also introduce the limit value of the indices of reducibility of parameter ideals of $M$.