Rings Over Which Flat Covers of Finitely Generated Modules are Projective

Abstract

In Bican et al. (2001), it is proved that all modules over an arbitrary ring have flat covers. In this article, we shall study rings over which flat covers of finitely generated modules are projective. We call a ring R right almost-perfect if every flat right R-module is projective relative to R. It turns out that a ring is right almost-perfect if and only if flat covers of finitely generated modules are projective. We shall show that the class of almost-perfect rings is properly between the class of perfect and semiperfect rings. We also outline some new characterizations of perfect rings. For example, we show that a ring R is right perfect if every finitely cogenerated right R-module has a projective cover.

Key Words:

• Almost-perfect ring
• Flat cover
• Perfect ring
• Projective cover
• Semiperfect ring

2000 Mathematics Subject Classification:

• 16D40
• 16L30

ACKNOWLEDGMENT

The authors would like to thank the referee for the valuable comments and suggestions.

Notes

Communicated by T. Albu.