Cotorsion Theories Involving Auslander and Bass Classes

Abstract

Let R be a commutative ring, 𝒞 be a semidualizing R-module. We show that the Auslander class 𝒜_𝒞(R) with respect to 𝒞 is the first left orthogonal class of some pure injective module M, that is, 𝒜_𝒞(R) =^⊥₁ M, and the Bass class ℬ_𝒞(R) is the first right orthogonal class of some G _𝒞-projective module N, that is, ℬ_𝒞(R) = N ^⊥₁ . As applications, we can see that (𝒜_𝒞(R), 𝒜_𝒞(R)^⊥) is a cotorsion theory generated by a set. Especially, we show that (^⊥ℬ_𝒞(R), ℬ_𝒞(R)) is a complete hereditary cotorsion theory cogenerated by a set.

Key Words:

• Auslander class
• Bass class
• Cotorsion theory
• (Pre)covering
• (Pre)enveloping
• Pure submodule
• Semidualizing module

2010 Mathematics Subject Classification:

• 13D02
• 13D07
• 13E15
• 16D10
• 18A20

ACKNOWLEDGMENTS

The authors would like to express their sincere thanks to the referee for his or her careful reading of the manuscript and helpful suggestions.

This research was partially supported by the National Natural Science Foundation of China (No. 10971090).

Notes

Communicated by I. Swanson.