# Almost Gorenstein rings arising from fiber products

## TL;DR

This paper characterizes when fiber products of Cohen-Macaulay local rings are almost Gorenstein, showing it occurs precisely when the component rings are also almost Gorenstein, thus advancing the understanding of Gorenstein properties in ring theory.

## Contribution

It provides a necessary and sufficient condition for fiber products of Cohen-Macaulay rings to be almost Gorenstein, extending the classification within the stratification of Cohen-Macaulay rings.

## Key findings

- Fiber product is almost Gorenstein if and only if the component rings are almost Gorenstein.
- The paper explores generalizations of Gorenstein properties in fiber products.
- Results contribute to the classification of Cohen-Macaulay rings based on Gorenstein-like properties.

## Abstract

The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local rings $R$, $S$ of the same dimension $d>0$ over a regular local ring $T$ with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are $R$ and $S$. Besides, the other generalizations of Gorenstein properties are also explored.