Elias Ideals

TL;DR

This paper studies Elias ideals in one-dimensional Cohen-Macaulay local rings, providing characterizations, criteria for identification, and connections to other ideal classes, with applications to canonical ideals and conductors.

Contribution

It introduces and characterizes Elias ideals, linking them to known ideal classes and offering criteria for their recognition in Cohen-Macaulay rings.

Findings

Elias ideals are characterized and their properties are established.

Criteria for identifying Elias ideals are provided.

Connections to Ulrich, trace, and integrally closed ideals are demonstrated.

Abstract

Let $(R, \mathfrak m)$ be a one dimensional local Cohen-Macaulay ring. An $\mathfrak m$-primary ideal $I$ of $R$ is Elias if the types of $I$ and of $R/I$ are equal. Canonical and principal ideals are Elias, and Elias ideals are closed under inclusion. We give multiple characterizations of Elias ideals and concrete criteria to identify them. We connect Elias ideals to other well-studied definitions: Ulrich, $\mathfrak m$-full, integrally closed, trace ideals, etc. Applications are given regarding canonical ideals, conductors and the Auslander index.