The set of trace ideals of curve singularities

TL;DR

This paper investigates the structure of trace ideals in one-dimensional local domains, providing criteria for finiteness and exploring properties related to integrally closed ideals, birational extensions, and reflexive modules, especially in numerical semigroup rings.

Contribution

It introduces new criteria for rings to have finitely many trace and reflexive ideals, connecting these properties with integrally closed ideals and specific ring constructions.

Findings

Criteria for rings with finitely many trace ideals.

Characterization of rings with finitely many reflexive ideals.

Analysis of trace ideals in numerical semigroup rings of non-gap four.

Abstract

This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to birational extensions, reflexive ideals, and reflexive Ulrich modules. Special attention is given in the case of numerical semigroup rings of non-gap four. We then obtain a criterion for a ring to have a finite number of reflexive ideals up to isomorphism. Non-domains arising from fiber products are also explored.