The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

TL;DR

This paper explores one-dimensional Cohen-Macaulay rings with minimal trace ideals of the canonical module, introducing the concept of far-flung Gorenstein rings and analyzing their properties and related algebraic structures.

Contribution

It introduces the class of far-flung Gorenstein rings characterized by minimal trace ideals and investigates their connections with endomorphism algebras and numerical semigroup rings.

Findings

Upper bound for multiplicity of far-flung Gorenstein numerical semigroup rings from Rohrbach problem

Characterization of reflexive modules over far-flung Gorenstein rings

Relation between trace ideals and Gorenstein properties in one-dimensional rings

Abstract

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.