On completely decomposable and separable modules over Pr\"ufer domains

TL;DR

This paper extends the theory of decomposable and separable modules from abelian groups to modules over h-local Pr"ufer domains, establishing new summand properties and a Pontryagin-Hill type theorem.

Contribution

It generalizes known results to a broader class of modules over Pr"ufer domains and proves a new Pontryagin-Hill type theorem for countable chains.

Findings

Summands of completely decomposable torsion-free modules are completely decomposable.

Summands of separable torsion-free modules are separable.

A Pontryagin-Hill type theorem is established for modules over h-local Pr"ufer domains.

Abstract

We generalize known results on summands of completely decomposable and separable torsion-free abelian groups to modules over h-local Pr\"ufer domains. Over such domains summands of completely decomposable torsion-free modules are again completely decomposable (Theorem 3.2) and summands of separable torsion-free modules are likewise separable (Theorem 4.2). In addition, a Pontryagin-Hill type theorem is established on countable chains of homogeneous completely decomposable modules over h-local Pr\"ufer domains.