On Divisible and Torsionfree Modules

Abstract

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.

Key Words:

• D-flat module
• D-injective module
• Divisible module
• P-coherent ring
• (Pre)Cover
• (Pre)Envelope
• Torsionfree module
• Warfield cotorsion module

2000 Mathematics Subject Classification:

• 16D50
• 16D40
• 16E10
• 18G10

ACKNOWLEDGMENTS

This research was partially supported by SRFDP (No. 20050284015), NSFC (No. 10771096), China Postdoctoral Science Foundation (No. 20060390926), Collegial Natural Science Research Program of Education Department of Jiangsu Province (No. 06KJB110033), and Jiangsu Planned Projects for Postdoctoral Research Funds (No. 060102IB). Science Research Fund of Nanjing Institute of Technology (KXJ07061), Jiangsu 333 Project, and Jiangsu Qinglan Project. The authors would like to thank the referee for the helpful comments and suggestions.

Notes

Communicated by A. Facchini.