On Covers and Envelopes in Some Functor Categories

Abstract

We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every left R-module M has an Ext-phantom preenvelope f: M → N with coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by the FP-injective objects.

Key Words:

• Cotorsion functor
• Cotorsion pair
• Ext-phantom
• Flat functor
• FP-injective functor
• FP-projective functor
• Phantom
• (Pre)envelope
• (Pre)cover
• Torsion theory

2010 Mathematics Subject Classification:

• 16B50
• 16D90
• 16E30
• 18A25
• 18G15
• 18G25

ACKNOWLEDGMENTS

This research was supported by NSFC (No. 11071111, 11171149), NSF of Jiangsu Province of China (No. BK2011068), Jiangsu 333 Project, Jiangsu Six Major Talents Peak Project. The author would like to thank the referee for the very helpful comments and suggestions.

Notes

Communicated by T. Albu.