On universal modules with pure embeddings

We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first-order theory with an infinite model extending the theory of R-modules and ${\mathbf{K}}^T=(Mod(T),{\le }_{\mathrm{pp}})$ (where ⩽_pp stands for “pure submodule”). Assume ${\mathbf{K}}^T$ has the joint embedding and amalgamation properties. If ${\lambda }^{|T|}=\lambda$ or $\forall \mu <\lambda ({\mu }^{|T|}<\lambda )$, then ${\mathbf{K}}^T$ has a universal model of cardinality λ. As a special case, we get a recent result of Shelah 28, 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 18, Question 4.25]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.