Cotorsion Theories Cogenerated by a Torsionfree Class

Abstract

Let R be a right perfect ring, and let (ℱ, 𝒞) be a cotorsion theory in the category of right R-modules ℳ _R . In this article, it is shown that every right R-module has a superfluous ℱ-cover if and only if there exists a torsion theory (𝒜, ℬ) such that (ℱ, 𝒞) is cogenerated by ℬ. It is also proved that if (𝒜, ℬ) is a cosplitting torsion theory, then (^⊥ℬ, (^⊥ℬ)^⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ℬ) is a centrally splitting torsion theory, then (^⊥ℬ, (^⊥ℬ)^⊥) is a hereditary and perfect cotorsion theory.

Key Words:

• Cotorsion theory
• Superfluous ℱ-cover
• Torsion theory

2000 Mathematics Subject Classification:

• 16E30
• 16S90
• 18E40

ACKNOWLEDGMENTS

This research was supported by Science Foundation for Young Teachers of Northeast Normal University (No. 09QNJJ003) and the Zhejiang Provincial Natural Science Foundation of China (No. Y6100173). The authors would like to thank the referee for the valuable comments and suggestions in shaping the article into its present form.

Notes

Communicated by T. Alln.