On some criteria for the balanced projectivity of modules over integral domains

TL;DR

This paper explores conditions under which unions of ascending chains of balanced-projective modules over integral domains remain balanced-projective, extending classical criteria and establishing new sufficient conditions.

Contribution

It introduces a countable union criterion for balanced-projectivity of torsion-free modules, generalizing Hill's freeness criterion for abelian groups.

Findings

Union of countable ascending chains of balanced-projective, pure submodules is balanced-projective.

Reduces the problem to the completely decomposable case.

Proves a Shelah-Eklof-type result and generalizes Auslander's lemma.

Abstract

Motivated by Hill's criterion of freeness for abelian groups, we investigate conditions under which unions of ascending chains of balanced-projective modules over integral domains are again balanced-projective. Our main result establishes that, in order for a torsion-free module to be balanced-projective, it is sufficient that it be the union of a countable, ascending chain of balanced-projective, pure submodules. The proof reduces to the completely decomposable case, and it hinges on the existence of suitable families of submodules of the links in the chain. A Shelah-Eklof-type result for the balanced projectivity of modules is proved in the way, and a generalization of Auslander's lemma is obtained as a corollary.