Cohen-Macaulay and Gorenstein properties under the amalgamated construction

TL;DR

This paper investigates how the Cohen-Macaulay and Gorenstein properties are preserved or characterized in the amalgamated construction of rings, expanding understanding of these properties in complex ring extensions.

Contribution

It provides new criteria and results for Cohen-Macaulay and Gorenstein properties in the amalgamation of rings along an ideal, a less explored construction.

Findings

Characterization of Cohen-Macaulay property in amalgamated rings

Conditions for Gorenstein and quasi-Gorenstein properties

Extension of classical properties to new ring constructions

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 with $J$ with respect to $f$. In this paper, among other things, we investigate the Cohen-Macaulay and (quasi-)Gorenstein properties on the ring $A\bowtie^fJ$.