Countably generated module
In mathematics, a module over a ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes from Kaplansky's theorem on [projective modules|Kaplansky's theorem], which states that a projective module is a direct [sum of modules|direct sum] of countably generated modules.
More generally, a module over a possibly non-commutative ring is projective if and only if it is flat, it is a direct sum of countably generated modules and it is a Mittag-Leffler module.