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REGULARITY RELATIVE TO A HEREDITARY TORSION THEORY FOR MODULES OVER A COMMUTATIVE RING

  • Qiao, Lei (School of Mathematical Sciences Sichuan Normal University) ;
  • Zuo, Kai (School of Mathematics Chengdu Normal University)
  • Received : 2021.12.28
  • Accepted : 2022.04.25
  • Published : 2022.07.01

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 Prüfer 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 ∞.

Keywords

Acknowledgement

The authors would like to thank Prof. Fanggui Wang and Dr. Mingzhao Chen for their helpful comments. The authors would also like to thank the referee for a careful reading of this manuscript and for correcting several errors.

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