Abstrakt
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 Prüfer domains of finite character with uncountable spectrum, and in particular, for the polynomial ring KTxU, where K is an uncountable field. One can even prescribe the 0-invariant of the union.
Our results generalize earlier work of Hill,and complement recent papers by Macías-Díaz, Fuchs, and Rangaswamy.