// Workers AI · dad joke modeWhy did the countably generated module go to therapy? It had a lot to account for.

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.
• 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.