On w_∞-projective modules and Krull domains

Abstract

Let R be a commutative ring with identity. In this paper, w_∞-projective modules are introduced and studied. It is shown that every R-module has a special w_∞-projective precover. As an application, it is proved that a domain R is a Krull domain if and only if every submodule of a w_∞-projective R-module is w_∞-projective. And we show that ${\mathcal{P}}_w^{{†}_{\infty }}⫋{\mathcal{W}}_{\infty }$ for any Krull domain R with ${\text{pd}}_{R}Q=2,$ where ${\mathcal{W}}_{\infty }$ denotes the class of all strong w-modules and ${\mathcal{P}}_w^{{†}_{\infty }}$ denotes the class of $\text{GV}$-torsionfree R-modules N with the property that ${\text{Ext}}_R^k(M,N)=0$ for all w-projective R-modules M and all integers $k\ge 1.$

Keywords:

• Krull domain
• w_∞-projective dimension
• w_∞-projective global dimension
• w_∞-projective module

2020 Mathematics Subject Classification::

• 13D05
• 13D07
• 13D30
• 13F05

Acknowledgments

The authors would like to thank the referee for the helpful comments and suggestions which substantially improved the paper.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Nos. 11861001, 11961050, and 12061001), and ABa teachers university (Nos. ASZ20-03 and ASA20-02).