The Milnor Number of One Dimensional Local Rings

TL;DR

This paper introduces an analogue of the Milnor number for one dimensional local rings, exploring its properties and connections to semigroups and ADE singularities.

Contribution

It defines a new invariant for one dimensional local rings and relates it to existing structures like semigroups and classical ADE singularities.

Findings

The Milnor number analogue satisfies properties similar to those in plane curve singularities.

Two new semigroup analogues are introduced and related to the Milnor number.

Connections between one dimensional rings of finite Cohen-Macaulay type and ADE singularities are established.

Abstract

In this paper we present an analogue of the Milnor number for one dimensional local ring, and we show that it satisfies analogous properties to those of the Milnor number of plane curves over a field. In addition, we present two analogues of the semi-group of values for a one dimensional ring and show how they relate to our Milnor number. Finally, we use these tools and techniques to show we can relate these semigroups of one dimensional rings of finite Cohen-Macaulay type to those of the classical ADE singularities.