#### Vol. 303, No. 1, 2019

 Recent Issues Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
Denoetherianizing Cohen–Macaulay rings

### László Fuchs and Bruce Olberding

Vol. 303 (2019), No. 1, 133–164
##### Abstract

We introduce a new class of commutative nonnoetherian rings, called $n$-subperfect rings, generalizing the almost perfect rings that have been studied recently by Fuchs and Salce. For an integer $n\ge 0$, the ring $R$ is said to be $n$-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 $ht\left(M\right)$-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.

##### Keywords
Perfect, subperfect, $n$-subperfect rings, regular sequence, unmixed, Cohen–Macaulay rings
##### Mathematical Subject Classification 2010
Primary: 13F99, 13H10
Secondary: 13C13