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.