Free algebras, universal models and Bass modules

TL;DR

This paper investigates when free structures of infinite rank in a variety exhibit strong model-theoretic universality properties, focusing on left $R$-modules and the class of flat modules. It develops a Bass-theoretic framework, starting from a descending chain of principal right ideals to form Bass modules and extends these ideas to a general theory of pure-projective modules via descending chains of pp formulas, establishing deep connections between ring-theoretic properties (notably left perfectness and coherence) and model-theoretic phenomena (universality, saturation, and total transcendence) for the theory $T$ of free modules. The authors prove that left perfect rings are precisely those for which the free module $R^{(\kappa)}$ is pure-universal among $\kappa$-generated flat $R$-modules and among $\kappa$-generated flat models of $T$, and they show that all models of $T$ are projective under these conditions, with a robust generalization to pure-projective contexts. These results yield a model-theoretic repro of Simson’s classical statements on pure-semisimple rings and Mittag-Leffler modules, and they provide a versatile toolkit (Bass theory) for analyzing definable subcategories, universality, and the interplay between algebraic structure and logical behavior in module categories.

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 $R^{(κ)}$ 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 $R^{(κ)}$ whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: $R^{(κ)}$ purely (equivalently, elementarily) embeds every $κ$-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.

Theorems & Definitions (33)

• Lemma 6
• Remark 7
• Proposition 8
• proof
• Remark 9
• Theorem 1
• proof
• Remark 10
• Example 11
• Theorem 2