Bass modules and embeddings into free modules

Abstract

We show that the free module of infinite rank $R^{\left(\kappa \right)}$ purely embeds every κ-generated flat left R-module iff R is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory T of $R^{\left(\kappa \right)}$ whose projectivity is equivalent to left perfectness, which allows to add a ‘stronger’ equivalent condition: $R^{\left(\kappa \right)}$ purely embeds every κ-generated flat left R-module which is a model of T.

We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a ‘Bass theory’ of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.