Rings with canonical reduction

TL;DR

This paper investigates one-dimensional Cohen-Macaulay local rings with a canonical ideal that reduces the maximal ideal, exploring their properties, relationships with almost Gorenstein rings, idealizations, and numerical semigroup rings.

Contribution

It introduces the concept of rings with canonical reduction and explores their connections to known classes like almost Gorenstein rings and numerical semigroup rings.

Findings

Rings with canonical reduction include almost Gorenstein rings.

Established relations between canonical reduction rings and idealizations.

Connected canonical reduction rings to numerical semigroup rings.

Abstract

The aim of this note is to study the class of one dimensional Cohen-Macaulay local rings, $(R, \mathfrak{m})$ say, possessing a canonical ideal $K$ which is a reduction of $\mathfrak{m}$. We call $R$ to have canonical reduction $K$. We show that the class of rings with canonical reductions contains the class of almost Gorenstein rings and establish its relation with rings obtained by idealizations and also with numerical semigroup rings.