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

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.