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
Departamento de Matemáticas y
Física
Universidad Autónoma de Aguascalientes
Avenida Universidad 940
Ciudad Universitaria
Aguascalientes, Ags. 20100
Mexico