Bass modules and embeddings into free modules
Anand Pillay, Philipp Rothmaler
TL;DR
This work characterizes when the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module, showing equivalence with left perfectness. It develops generalized Bass modules from descending chains of pp formulas and a Bass theory of pure-projective modules, linking model-theoretic theories of free modules to module-theoretic purity. The authors establish a network of equivalences among dcc conditions, Mittag-Leffler properties of $\mathcal{C}$-models of $T_{\mathcal{A}}$, and pure-projectivity, and use Bass-modules to produce models of these theories; non-stabilizing chains yield non-pure-projective instances. The framework rederives classic results (e.g., Simson) and extends to cyclically presented and RD-projective modules, illuminating when flat modules decompose into pure-free or projective components and clarifying the role of left perfectness in the structure of pure embeddings.
Abstract
We show 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 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. 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).