# Countably generated module

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{{Short description|Module generated by a countable subset}}
In [mathematics](/source/mathematics), a [module](/source/module_(mathematics)) over a (not necessarily [commutative](/source/commutative_ring)) [ring](/source/ring_(mathematics)) is '''countably generated''' if it is [generated](/source/generating_set_of_a_module) as a module by a [countable](/source/countable) subset. The importance of the notion comes from [Kaplansky's theorem](/source/Kaplansky's_theorem_on_projective_modules) (Kaplansky 1958), which states that a [projective module](/source/projective_module) is a [direct sum](/source/direct_sum_of_modules) of countably generated modules.

More generally, a module over a possibly [non-commutative](/source/non-commutative_ring) ring is projective [if and only if](/source/if_and_only_if) (i) it is [flat](/source/flat_module), (ii) it is a direct sum of countably generated modules and (iii) it is a [Mittag-Leffler module](/source/Mittag-Leffler_module). (Bazzoni–Stovicek)

==References==
* {{cite journal|jstor=1970252|author-link=Irving Kaplansky|title=Projective Modules|journal=Annals of Mathematics|volume=68|issue=2|pages=372–377|last1=Kaplansky|first1=Irving|year=1958|doi=10.2307/1970252|hdl=10338.dmlcz/101124|hdl-access=free}}
* {{cite journal|arxiv=1007.4977|doi=10.1090/S0002-9939-2011-11070-0|doi-access=free|title=Flat Mittag-Leffler modules over countable rings|journal=[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)|volume=140|issue=5|pages=1527–1533|year=2012|last1=Bazzoni|first1=Silvana|last2=Šťovíček|first2=Jan}}

Category:Module theory

{{linear-algebra-stub}}

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