Cohen-Macaulay properties under the amalgamated construction

TL;DR

This paper investigates the Cohen-Macaulay property in the context of amalgamated rings, generalizing known results and extending the concept to non-Noetherian cases through the amalgamation construction.

Contribution

It introduces a study of Cohen-Macaulay properties in amalgamated rings, generalizing classical results and exploring the property beyond Noetherian rings.

Findings

Generalizes Cohen-Macaulay conditions to amalgamated rings

Extends results from Nagata's idealization to broader contexts

Provides criteria for Cohen-Macaulayness in amalgamations

Abstract

Let $A$ and $B$ be commutative rings with unity, $f:A\to B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $A\bowtie^fJ:=\{(a,f(a)+j)|a\in A$ and $j\in J\}$ of $A\times B$ is called the amalgamation of $A$ with $B$ along $J$ with respect to $f$. In this paper, we study the property of Cohen-Macaulay in the sense of ideals which was introduced by Asgharzadeh and Tousi, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $A\bowtie^fJ$. Among other things, we obtain a generalization of the well-known result that when the Nagata's idealization is Cohen-Macaulay.