Submission date: 1 March 2019

Abstract:

We show that certain classes of modules have universal models with respect to pure embeddings.

Theorem. Let R be a ring, T a first-order theory with an infinite model extending the theory of R-modules and K^T=(Mod(T), ≤_{pp}) (where ≤_{pp} stands for pure submodule). Assume K^T has joint embedding and that pure-injective modules are amalgamation bases. If λ^{|T|}=λ or ∀ μ < λ( μ^{|T|} < λ), then K^T has a universal model of cardinality λ.
As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings.
We begin the study of limit models for classes of R-modules with joint embedding and amalgamation. As a by-product of this study, we characterize limit models of countable cofinality in the class of torsion-free abelian groups with pure embeddings, answering Question 4.25 of [Maz].
Theorem. If G is a (λ, ω)-limit model in the class of torsion-free groups with pure embeddings, then G ≅ ℚ^{(λ)} ⊕ ∏_{p} \overline{ℤ_{(p)}^{(λ)}}^{(ℵ_0)}.
As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

Mathematics Subject Classification: 03C45, 03C48, 03C60, 13L05, 16D10

Keywords and phrases:

Full text arXiv 1903.00414: pdf, ps.