A generalization of the Pontryagin-Hill theorems to projective modules over Pr\"ufer domains

TL;DR

This paper extends classical criteria for freeness in abelian groups to projective modules over Pr"ufer domains, showing that unions of certain ascending chains of projective modules remain projective, generalizing Hill's theorem.

Contribution

It generalizes Hill's theorem to projective modules over Pr"ufer domains with countably many maximal ideals, establishing new conditions for projectivity of unions of modules.

Findings

Unions of countable ascending chains of projective, pure submodules are projective over Pr"ufer domains.

Extended Hill's theorem applies to modules over arbitrary rings and domains.

Provides new criteria for projectivity in module theory.

Abstract

Motivated by the Pontryagin-Hill criteria of freeness for abelian groups, we investigate conditions under which unions of ascending chains of projective modules are again projective. Several extensions of these criteria are proved for modules over arbitrary rings and domains, including a genuine generalization of Hill's theorem for projective modules over Pr\"{u}fer domains with a countable number of maximal ideals. More precisely, we prove that, over such domains, modules which are unions of countable ascending chains of projective, pure submodules are likewise projective.