Free algebras, universal models and Bass modules

2024/07/17 by Pillay, Anand, Rothmaler, Philipp
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2407.15864

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.

Related

• Bass modules and embeddings into free modules
• Remarks on theories of free algebras and modules
• On universal modules with pure embeddings
• A relative version of Bass' theorem about finite-dimensional algebras
• Almost free modules and Mittag--Leffler conditions