Quasi-socle ideals in Gorenstein numerical semigroup rings

TL;DR

This paper investigates quasi-socle ideals in Gorenstein numerical semigroup rings, focusing on their integrality over parameter ideals and Cohen-Macaulay properties of their associated graded rings, with various examples illustrating complex behaviors.

Contribution

It provides new insights into the structure and properties of quasi-socle ideals in Gorenstein numerical semigroup rings, including conditions for integrality and Cohen-Macaulayness.

Findings

Examples of quasi-socle ideals with diverse properties.

Conditions under which the associated graded ring is Cohen-Macaulay.

Instances where the ideal is or is not integral over the parameter ideal.

Abstract

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal ideal in the base local ring, and $q \geq 1$ is an integer. The problems of when $I$ is integral over $Q$ and of when the associated graded ring $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are studied. The problems are rather wild; examples are given.