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

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.