Ulrich ideals in numerical semigroup rings of small multiplicity

TL;DR

This paper investigates Ulrich ideals in numerical semigroup rings with small multiplicity, revealing their structure in symmetric cases and exploring complexities as multiplicity increases, especially beyond three.

Contribution

It provides explicit descriptions of Ulrich ideals in semigroup rings of multiplicity up to 3 and discusses the challenges in higher multiplicities, notably multiplicity 4.

Findings

Ulrich ideals in non-symmetric three-generated semigroup rings are absent due to Golod properties.

Explicit generators for Ulrich ideals are identified in symmetric cases with multiplicity up to 3.

Determining Ulrich ideals becomes complex for semigroup rings with multiplicity greater than 3.

Abstract

Ulrich ideals in numerical semigroup rings of small multiplicity are studied. If the semigroups are three-generated but not symmetric, the semigroup rings are Golod, since the Betti numbers of the residue class fields of the semigroup rings form an arithmetic progression; therefore, these semigroup rings are G-regular, possessing no Ulrich ideals. Nevertheless, even in the three-generated case, the situation is different, when the semigroups are symmetric. We shall explore this phenomenon, describing an explicit system of generators, that is the normal form of generators, for the Ulrich ideals in the numerical semigroup rings of multiplicity at most 3. As the multiplicity is greater than $3$, in general the task of determining all the Ulrich ideals seems formidable, which we shall experience, analyzing one of the simplest examples of semigroup rings of multiplicity $4$.