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)
See also
- Countable ring
- Countable dimension
References
- I. Kaplansky, Projective modules, http://www.jstor.org/stable/1970252.
- S. Bazzoni, J. Stovicek, Flat Mittag-Leffler modules over countable rings, arXiv:1007.4977v2 [math.RA].
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