countably generated module
In mathematics, a module over a (not necessarily commutative) 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 (Kaplansky 1958), which states that a projective module is a direct sum of countably generated modules.
More generally, a module over a possibly non-commutative ring is projective if and only if (i) it is flat, (ii) it is a direct sum of countably generated modules and (iii) it is a 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|volume=140|issue=5|pages=1527–1533|year=2012|last1=Bazzoni|first1=Silvana|last2=Šťovíček|first2=Jan}}
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