The index of a numerical semigroup ring

TL;DR

This paper computes the generalized Loewy length (Auslander's index) for certain numerical semigroup rings, explores Ding's inequality, and extends index calculations to rings with higher embedding dimensions.

Contribution

It provides explicit formulas for the index of complete intersection numerical semigroup rings and demonstrates the unboundedness of Ding's inequality for embedding dimension three.

Findings

The generalized Loewy length is expressed in terms of minimal generators.

Ding's inequality can be arbitrarily violated for rings with embedding dimension three.

Index calculations are extended to rings with higher embedding dimensions.

Abstract

Let $R=k[|t^a,t^b,t^c|]$ be a complete intersection numerical semigroup ring over an infinite field $k$, where $a,b,c\in\BN$. The generalized Loewy length, which is Auslander's index in this case, is computed in terms of the minimal generators of the semigroup: $a,b$ and $c$. Examples provided show that the left hand side of Ding's inequality $\mult(R)-\inde(R)-\codim(R)+1\geq 0$ can be made arbitrarily large for rings $R$ with $\edim(R)=3$ . The index of a complete intersection numerical semigroup ring with embedding dimension greater than three is also computed.