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

• Kaplansky, Irving (1958). "Projective Modules". Annals of Mathematics. 68 (2): 372–377. doi:10.2307/1970252. hdl:10338.dmlcz/101124. JSTOR 1970252.372-377&rft.date=1958&rft_id=info:hdl/10338.dmlcz/101124&rft_id=https://www.jstor.org/stable/1970252#id-name=JSTOR&rft_id=info:doi/10.2307/1970252&rft.aulast=Kaplansky&rft.aufirst=Irving&rfr_id=info:sid/en.wikipedia.org:Countably generated module" class="Z3988">
• Bazzoni, Silvana; Šťovíček, Jan (2012). "Flat Mittag-Leffler modules over countable rings". Proceedings of the American Mathematical Society. 140 (5): 1527–1533. arXiv:1007.4977. doi:10.1090/S0002-9939-2011-11070-0.1527-1533&rft.date=2012&rft_id=info:arxiv/1007.4977&rft_id=info:doi/10.1090/S0002-9939-2011-11070-0&rft.aulast=Bazzoni&rft.aufirst=Silvana&rft.au=Šťovíček, Jan&rft_id=https://doi.org/10.1090%2FS0002-9939-2011-11070-0&rfr_id=info:sid/en.wikipedia.org:Countably generated module" class="Z3988">