Envelopes of Covers and Covers of Envelopes^#

^#Communicated by R. Wisbauer.

Let ℒ be a class of right modules, whenever covers and envelopes exist, we show that if ℒ is closed under extensions and direct summands, then P(ℒ), the class of all ℒ-projective right modules, is closed under taking ℒ-envelopes and ℒ is closed under taking ℒ-projective covers; and that if P(ℒ) is resolving, then the P(ℒ)-injective envelope of the ℒ-projective cover of a module is isomorphic to the ℒ-projective cover of the P(ℒ)-injective envelope of this module. As a corollary, we derive the result of Xu (1996, Theorem 3.4.8) that the flat cover of the cotorsion envelope of M is isomorphic to the cotorsion envelope of the flat cover of M.

Key Words:

• Cover
• Envelope
• Pullback
• Pushout

Mathematics Subject Classification:

• 16D10
• 16E40

ACKNOWLEDGMENTS

The author wishes to thank Professors Wenting Tong and Weimin Xue for many valuable suggestions and helpful communications. He is indebted to the referee for several comments that improved the article. This research is supported by the Natural Science Foundation of China (No. A0324656) and the Fumiao Foundation of Fujian Normal University (No. F029).

Notes

^# Communicated by R. Wisbauer.