The Goto numbers of parameter ideals

TL;DR

This paper investigates the Goto numbers of parameter ideals in Noetherian local rings, focusing on one-dimensional cases and numerical semigroup rings, to understand their behavior and properties.

Contribution

It provides new insights into the values of Goto numbers for parameter ideals, especially in one-dimensional and numerical semigroup ring contexts.

Findings

Characterization of Goto numbers in one-dimensional rings

Analysis of Goto numbers in numerical semigroup rings

Identification of conditions affecting Goto number values

Abstract

Let Q be a parameter ideal of a Noetherian local ring (R,m). The Goto number g(Q) of Q is the largest integer g such that Q:m^g is integral over Q. We examine the values of g(Q) as Q varies over the parameter ideals of R. We concentrate mainly on the case where dim R = 1, and many of our results concern parameter ideals of a numerical semigroup ring.