Regularity relative to a hereditary torsion theory for modules over a commutative ring
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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 $\infty$.
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 $\infty$.
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