Strong submodules of almost projective modules

The structure of almost projective modules can be better understood in the case when the following Condition (P) holds: The union of each countable pure chain of projective modules is projective. We prove this condition, and its generalization to pure-projective modules, for all countable rings, using the new notion of a strong submodule of the union. However, we also show that Condition (P) fails for all Priifer domains of finite character with uncountable spectrum, and in particular, for the polynomial ring K[x], where K is an uncountable field. One can even prescribe the Γ-invariant of the union. Our results generalize earlier work of Hill, and complement recent papers by Macias-Diaz, Fuchs, and Rangaswamy.