When is $\mathfrak{m}:\mathfrak{m}$ an almost Gorenstein ring?

TL;DR

This paper characterizes when a one-dimensional Cohen-Macaulay local ring is almost Gorenstein based on the properties of its maximal ideal as a canonical module, introducing new concepts like almost canonical ideals and gAGL rings.

Contribution

It introduces the notions of almost canonical ideals and gAGL rings, providing new characterizations of almost Gorenstein rings and generalizing previous results.

Findings

Characterization of almost Gorenstein rings via the maximal ideal as a canonical module.

Introduction of almost canonical ideals and gAGL rings.

Extension of known results to rings with minimal multiplicity.

Abstract

Given a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak{m},k)$, we prove that it is almost Gorenstein if and only if $\mathfrak{m}$ is a canonical module of the ring $\mathfrak{m}:\mathfrak{m}$. Then, we generalize this result by introducing the notions of almost canonical ideal and gAGL ring and by proving that $R$ is gAGL if and only if $\mathfrak{m}$ is an almost canonical ideal of $\mathfrak{m}:\mathfrak{m}$. We use this fact to characterize when the ring $\mathfrak{m}:\mathfrak{m}$ is almost Gorenstein, provided that $R$ has minimal multiplicity. This is a generalization of a result proved by Chau, Goto, Kumashiro, and Matsuoka in the case in which $\mathfrak{m}:\mathfrak{m}$ is local and its residue field is isomorphic to $k$.