Free algebras, universal models and Bass modules

TL;DR

This paper explores the model-theoretic properties of free modules of infinite rank over rings, establishing conditions for universality and saturation, and introduces a Bass theory for pure-projective modules with applications to classical results.

Contribution

It introduces a Bass theory of pure-projective modules and links model-theoretic properties of free modules to ring-theoretic conditions like left perfectness.

Findings

Free modules of infinite rank embed all flat modules iff the ring is left perfect.

Constructs models where projectivity aligns with ring properties.

Reproves classical results on pure-semisimple rings using Bass modules.

Abstract

We investigate the question of when free structures of infinite rank (in a variety) possess model-theoretic properties like categoricity in higher power, saturation, or universality. Concentrating on left $R$-modules we show, among other things, 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 (equivalently, elementarily) embeds every $\kappa$-generated flat left $R$-module which is a model of $T$. In addition, 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 reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.