When are trace ideals finite?

TL;DR

This paper characterizes Noetherian local rings with finitely many trace ideals, establishing their dimension bounds and providing conditions related to Cohen-Macaulay rings and canonical modules.

Contribution

It proves dimension bounds for rings with finitely many trace ideals and links trace ideals to canonical modules and endomorphism algebras in specific cases.

Findings

Rings with finitely many trace ideals have dimension at most two.

If the integral closure of $R/H$ is equi-dimensional, then the ring has dimension at most one.

A necessary condition involving the canonical module's value set for finiteness of trace ideals.

Abstract

In this paper, we study Noetherian local rings $R$ having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral closure of $R/H$, where $H$ is the zeroth local cohomology, is equi-dimensional, then the dimension of $R$ is at most one. In the one-dimensional case, we can reduce to the situation that rings are Cohen-Macaulay. Then, we give a necessary condition to have a finite number of trace ideals in terms of the value set obtained by the canonical module. We also gave the correspondence between trace ideals of $R$ and those of the endomorphism algebra of the maximal ideal of $R$ when $R$ has minimal multiplicity.