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Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2022; 59(4): 821-841

Published online July 1, 2022 https://doi.org/10.4134/JKMS.j210774

Copyright © The Korean Mathematical Society.

Regularity relative to a hereditary torsion theory for modules over a commutative ring

Lei Qiao, Kai Zuo

Sichuan Normal University; Chengdu Normal University

Abstract

In this paper, we introduce and study regular rings relative to the hereditary torsion theory $w$ (a special case of a well-centered torsion theory over a commutative ring), called $w$-regular rings. We focus mainly on the $w$-regularity for $w$-coherent rings and $w$-Noetherian rings. In particular, it is shown that the $w$-coherent $w$-regular domains are exactly the Pr\"ufer $v$-multiplication domains and that an integral domain is $w$-Noetherian and $w$-regular if and only if it is a Krull domain. We also prove the $w$-analogue of the global version of the Serre--Auslander-Buchsbaum Theorem. Among other things, we show that every $w$-Noetherian $w$-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak $w$-projective dimension of a $w$-Noetherian ring is 0, 1, or $\infty$.

Keywords: Weak $w$-projective module, weak $w$-projective dimension, $w$-regular ring, $w$-coherent ring, $w$-Noetherian ring

MSC numbers: 13D05, 13D30, 13A15

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