# On universal modules with pure embeddings

## TL;DR

This paper investigates the existence of universal modules with pure embeddings over rings, establishing conditions for their existence and analyzing the structure of limit models, thus advancing the understanding of module theory in model-theoretic terms.

## Contribution

It proves the existence of universal models for classes of modules with pure embeddings under certain cardinality conditions and studies the structure of limit models, connecting model theory and algebra.

## Key findings

- Universal models exist under specific cardinality conditions.
- Limit models with long cofinality chains are pure-injective.
- Characterization of limit models with countable cofinality.

## 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 $K^T=(Mod(T), \leq_{pp})$ (where $\leq_{pp}$ stands for pure submodule). Assume $K^T$ has joint embedding and amalgamation. 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. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer Question 4.25 of [Maz]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.