Bass modules and embeddings into free modules

TL;DR

This paper characterizes when free modules of infinite rank can embed all flat modules using Bass modules, linking this property to the left perfectness of the ring, and develops a Bass theory for pure-projective modules.

Contribution

It introduces a new Bass theory of pure-projective modules and connects embedding properties of free modules to ring perfectness, extending model-theoretic methods.

Findings

Pure embedding of flat modules characterizes left perfect rings.

Constructs a model-theoretic Bass module for arbitrary chains of pp formulas.

Reproves classical results on pure-semisimple rings and Mittag-Leffler modules.

Abstract

We show that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-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 $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a `stronger' equivalent condition: $R^{(\kappa)}$ purely embeds every $\kappa$-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. This paper is a condensed version, solely about modules, of our larger work arXiv:2407.15864, with two new results added about cyclically presented modules (Cor.14) and finitely presented cyclic modules (Rem.15).