When Are Covers Relative to Two Classes Isomorphic

Abstract

Let ℒ₁ ⊇ ℒ₂ be classes of right R-modules, we introduce ℒ₁-covering modules and projective modules relative to ℒ₂ to characterize the relations between the existences of ℒ₁-covers, ℒ₂-covers and the classes ℒ₁, ℒ₂. As corollaries, every module in ℱ _n is an ℱ-covering module if and only if every flat cover is an ℱ _n -cover for each right module if and only if R is a von Neumann regular ring whenever wD(R) < ∞; every flat right R-module is projective relative to 𝒫 if and only if every flat cover is a projective cover for each right module if and only if R is right perfect.

Key Words:

• Cover
• Envelope
• ℒ-Covering
• ℒ-Projective

2000 Mathematics Subject Classification:

• 16D10
• 16E40

ACKNOWLEDGMENTS

This research is supported by the Natural Science Foundation of Fujian Province (2009J01003), the Scientific Research Foundation of Fujian Provincial Department of Science and Technology (2007F5038), and the Foundation of Fujian Normal University (2008100209, 09A004).

Notes

Communicated by T. Albu.