# A note on Non-Noetherian Cohen-Macaulay rings

## TL;DR

This paper investigates the Cohen-Macaulay property in non-Noetherian rings, demonstrating limitations of existing theorems and establishing new results for valuation domains.

## Contribution

It shows Hochster's theorem does not extend to non-Noetherian rings and proves that polynomial extensions of certain valuation domains are Cohen-Macaulay.

## Key findings

- Hochster's theorem does not hold in non-Noetherian cases
- Valuation domain polynomial extensions are Cohen-Macaulay
- Provides new insights into Cohen-Macaulayness beyond Noetherian rings

## Abstract

In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster's celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that for any valuation domain $V$ of finite Krull dimension, $V[x]$ is Cohen-Macaulay in the sense of Hamilton-Marley.