Relatively flat modules on ring extensions

Abstract

A ring extension is a ring homomorphism preserving identities. In this article, we establish the relationship among relatively flat modules, relatively projective modules and relatively injective modules on ring extensions. In particular, we prove that relatively projective modules are relatively flat, and that finitely presented and relatively flat modules are relatively projective. Moreover, we give a series of equivalent conditions of relatively flat modules on ring extensions from different perspectives. As applications, we prove that relatively flat modules are closed under extensions with respect to relative exact sequences, and obtain a necessary and sufficient condition for relative factor modules of relatively flat modules to be relatively flat.

Keywords:

• Relatively flat modules
• relatively injective modules
• relatively projective modules
• ring extensions

2020 Mathematics Subject Classification:

• 18E30
• 16D10
• 18G25

Acknowledgments

The authors thank the referee for helpful comments and suggestions.

Additional information

Funding

The research work is partially supported by the Guangxi Natural Science Foundation Project (No. 2022GXNSFAA035532).