When is $R \ltimes I$ an almost Gorenstein local ring?

TL;DR

This paper provides a complete characterization of when the idealization of an ideal over a Gorenstein local ring results in an almost Gorenstein local ring, focusing on the Cohen-Macaulay and Gorenstein properties.

Contribution

It offers a complete criterion for the almost Gorenstein property of the idealization of an ideal in a Gorenstein local ring.

Findings

Characterization of when $R owtie I$ is almost Gorenstein

Conditions relating Cohen-Macaulay and Gorenstein properties

Complete answer to the idealization question

Abstract

Let $(R, \mathfrak{m}) $ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a Cohen-Macaulay ring of dimension $d$. There is given a complete answer to the question of when the idealization $A = R \ltimes I$ of $I$ over $R$ is an almost Gorenstein local ring.