#### 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 $n\ge 0$, 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.

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Perfect, subperfect, $n$-subperfect rings, regular sequence, unmixed, Cohen–Macaulay rings