A new characterization of Cohen-Macaulay rings

TL;DR

This paper introduces a novel way to characterize Cohen-Macaulay local rings, linking Gorenstein rings to the irreducibility of parameter ideals, thereby deepening the understanding of their structural properties.

Contribution

It provides a new characterization of Cohen-Macaulay rings and establishes a criterion for Gorenstein rings based on parameter ideal irreducibility.

Findings

A local ring is Gorenstein if and only if all its parameter ideals are irreducible.

New characterization of Cohen-Macaulay rings based on ring properties.

Deeper insight into the structure of Gorenstein and Cohen-Macaulay rings.

Abstract

The purpose of this article is to provide a new characterization of Cohen-Macaulay local rings. As a consequence we deduce that a local (Noetherian) ring $R$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible.