Reducibility of parameter ideals in low powers of the maximal ideal

TL;DR

This paper investigates the conditions under which parameter ideals in low powers of the maximal ideal indicate a Gorenstein property in local rings, providing bounds based on specific systems of parameters.

Contribution

It introduces new upper bounds for the integer ll that determine Gorenstein rings via irreducible parameter ideals in low powers of the maximal ideal.

Findings

Upper bounds for ll depend on systems of parameters in low powers of rm.

Characterization of Gorenstein rings through irreducible parameter ideals.

Connections between parameter ideals and ring properties in low powers.

Abstract

A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer $\ell$ (depending on $R$) such that $R$ is Gorenstein if and only if there exists an irreducible parameter ideal contained in $\mathfrak{m}^\ell$. We give upper bounds for $\ell$ that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.