Lazard’s theorem for S-pure flat modules

ABSTRACT

Let R be an associative ring with identity. For a given class 𝒮 of finitely presented left (respectively right) R-modules containing R, we present a complete characterization of 𝒮-pure injective modules and 𝒮-pure flat modules. Consider that 𝒮 is a class of (R,R)-bimodules containing R with the following property: every element of 𝒮 is a finitely presented left and right R-module. We give a necessary and sufficient condition for 𝒮 to have Lazard’s theorem, and then we present our desired Lazard’s theorem.

KEYWORDS:

• Auslander transpose
• direct limit
• 𝒮-pure flat module
• 𝒮-pure injective module
• 𝒮-pure projective module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D80
• Secondary: 16D90
• 18A30
• 18E05
• 16D20

Acknowledgments

The author would like to express her gratitude to Professor Henrik Holm for his technical comments and the referee for her/his many constructive comments and valuable suggestions which improved this paper.