A note on the direct limit of increasing sequences of completely decomposable modules over integral domains

TL;DR

This paper generalizes Hill's criterion of freeness by establishing conditions under which the union of an increasing sequence of pure, completely decomposable modules over domains remains completely decomposable, emphasizing the role of purity.

Contribution

It introduces a new criterion for complete decomposability of modules as unions of pure, ascending chains, extending previous results in module theory.

Findings

Union of countable ascending chains of pure, completely decomposable modules are completely decomposable.

Purity of submodules is essential for the main result.

Generalizes Hill's criterion of freeness to modules over domains.

Abstract

In this note, we establish conditions under which the union of an increasing sequence of completely decomposable modules over domains are again completely decomposable. In our investigation, the condition of purity of modules is crucial. In fact, the main result reported in this work states that a module is completely decomposable when it is the union of a countable, ascending chain of completely decomposable, pure submodules, providing thus a generalization of Hill's criterion of freeness from abelian group theory.