A generalization of the Pontryagin–Hill theorems to projective modules over Prüfer domains

Jorge Macías-Díaz

Vol. 246 (2010), No. 2, 391–405

DOI: 10.2140/pjm.2010.246.391

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. We prove several extensions of these criteria for modules over arbitrary rings and domains, including a genuine generalization of Hill’s theorem for projective modules over Prüfer domains with a countable number of maximal ideals. More precisely, we prove that, over such domains, modules that are unions of countable ascending chains of projective, pure submodules are likewise projective.

Keywords

Pontryagin–Hill theorems, projective modules, Prüfer domains, G(ℵ₀)-families of submodules, pure submodules, relatively divisible submodules

Mathematical Subject Classification 2000

Primary: 13C05, 13C10

Secondary: 16D40, 13F05

Milestones

Received: 1 June 2009

Revised: 31 August 2009

Accepted: 13 October 2009

Published: 1 June 2010