On the unions of ascending chains of direct sums of ideals of h-local Pr\"{u}fer domains

TL;DR

This paper generalizes classical theorems to characterize when unions of ascending chains of modules, specifically direct sums of ideals in h-local Prüfer domains, are themselves direct sums of ideals, focusing on torsion-free modules over Dedekind domains.

Contribution

It extends Pontryagin-Hill type theorems to modules over h-local Prüfer domains, providing new conditions for unions of ascending chains to be direct sums of ideals.

Findings

Unions of ascending chains of pure submodules are isomorphic to direct sums of ideals.

Modules over Dedekind domains with countably many maximal ideals can be expressed as direct sums of ideals.

Generalizations of classical theorems for modules over specific integral domains.

Abstract

In this work, we investigate conditions under which unions of ascending chains of modules which are isomorphic to direct sums of ideals of an integral domain are again isomorphic to direct sums of ideals. We obtain generalizations of the Pontryagin-Hill theorems for modules which are direct sums of ideals of h-local Pr\"{u}fer domains. Particularly, we prove that a torsion-free module over a Dedekind domain with a countable number of maximal ideals is isomorphic to a direct sum of ideals if it is the union of a countable ascending chain of pure submodules which are isomorphic to direct sums of ideals.