Kaplansky's theorem on projective modules

Remarks

Part 1 is proved by a transfinite construction that builds a projective module up from countably generated direct summands. For part 2 one reduces to countably generated modules by part 1. A finitely generated projective module over a local ring is free by Nakayama's lemma. In the countably generated case, an inductive argument splits off free summands one element at a time. The point of part 2 is that no finiteness hypothesis is needed. Example: every projective module over the local ring $\Z_{(p)}$ is free. The local hypothesis is necessary. Over $\Z/6\Z \cong \Z/2\Z \times \Z/3\Z$, the module $\Z/2\Z$ is projective, since it is a direct summand of $\Z/6\Z$. It is not free, since it has $2$ elements while a nonzero free module has at least $6$.

Going deeper

For a student

For finitely generated modules, part 2 is a standard consequence of Nakayama's lemma; Kaplansky's contribution is the infinitely generated case. Part 1 is a general reduction used in many other questions about large projective modules. Over semiperfect rings a related result holds: every finitely generated projective module is a direct sum of modules $Re$ with $e$ an idempotent such that $eRe$ is local.