Vol. 303, No. 1, 2019

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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 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 RI 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 RM 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.

Keywords
Perfect, subperfect, $n$-subperfect rings, regular sequence, unmixed, Cohen–Macaulay rings
Mathematical Subject Classification 2010
Primary: 13F99, 13H10
Secondary: 13C13
Milestones
Received: 6 January 2018
Revised: 18 April 2019
Accepted: 28 April 2019
Published: 21 December 2019
Authors
László Fuchs
Department of Mathematics
Tulane University
New Orleans, LA
United States
Bruce Olberding
Department of Mathematical Sciences
New Mexico State University
PO Box 30001, MSC MB
Las Cruces, NM
United States