On universal modules with pure embeddings

Thomas G. Kucera, Marcos Mazari-Armida

Published 2019-03-01Version 1

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 $K^T=(Mod(T), \leq_{pp})$ (where $\leq_{pp}$ stands for pure submodule). Assume $K^T$ has joint embedding and that pure-injective modules are amalgamation bases. If $\lambda^{|T|}=\lambda$ or $\forall \mu < \lambda( \mu^{|T|} < \lambda)$, then $K^T$ has a universal model of cardinality $\lambda$. 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 $(\lambda, \omega)$-limit model in the class of torsion-free groups with pure embeddings, then $G \cong \mathbb{Q}^{(\lambda)} \oplus \prod_{p} \overline{\mathbb{Z}_{(p)}^{(\lambda)}}^{(\aleph_0)}$. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.