PCF and Abelian Groups

TL;DR

This paper explores pcf theory's role in constructing abelian groups with trivial duals, extending to R-modules, and demonstrates the existence of highly free structures under certain cardinal arithmetic conditions.

Contribution

It introduces new pcf-based methods for constructing abelian groups and R-modules with specific properties, broadening previous results and applications.

Findings

Existence of aleph_omega-free abelian groups with trivial duals

Development of pcf techniques for constructing free structures

Extension of results from abelian groups to R-modules

Abstract

We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that "almost always" there are F subseteq lambda^kappa which are quite free and has black boxes. The "almost always" means that there are strong restrictions on cardinal arithmetic if the universe fails this, the restrictions are "everywhere". Also we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer.