De-noetherizing Cohen-Macaulay rings

TL;DR

This paper introduces a new class of non-noetherian rings called n-subperfect rings, generalizing Cohen-Macaulay rings, and explores their properties and relation to classical Cohen-Macaulay rings.

Contribution

It defines n-subperfect rings and extended Cohen-Macaulay rings, extending Cohen-Macaulay theory to non-noetherian contexts with new structural insights.

Findings

n-subperfect rings generalize almost perfect rings

Extended Cohen-Macaulay rings have noetherian prime spectrum

Properties similar to classical Cohen-Macaulay rings are established

Abstract

We introduce a new class of commutative {non-noetherian} rings, called $n$-subperfect rings, generalizing the almost perfect rings that have been studied recently by Fuchs-Salce. For an integer $n \ge 0$, the ring $R$ is $n$-subperfect if every maximal regular sequence in $R$ has length $n$ and the total ring of quotients of $R/I$ for any ideal $I$ generated by a regular sequence is a perfect ring in the sense of Bass. We define an extended Cohen-Macaulay ring as a commutative ring $R$ that has noetherian prime spectrum and each localization $R_M$ at a maximal ideal $M$ is ht($M$)-subperfect. In the noetherian case, these are precisely the classical Cohen-Macaulay rings. Several relevant properties are proved reminiscent of those shared by Cohen-Macaulay rings.