De-noetherizing Cohen-Macaulay rings

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.