This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, Eisenbud, Bourbaki...) which I skipped as a little too complicated the first $N$ times around.
Recently I got the chance to read and understand I. Kaplansky's big theorem on projective modules, i.e., that a(n even infinitely generated) projective module over a local ring is free. En route to establishing this, he proves another result which is interesting but rather technical:
Theorem (Kaplansky, 1958): Every projective module is a direct sum of countably generated projective submodules.
For my take on this result, see $\S 3.10$ of these notes. In particular, it raises two natural questions:
Question 1: Is there a ring $R$ and an $R$-module $M$ which is not a direct sum of countably generated submodules?
Question 2: Is there a ring $R$ and a projective $R$-module $P$ which is not a direct sum of finitely generated submodules?
I was able to look up that the answer to Question 1 is "yes". In particular, I found work of L. Fuchs which says that for every infinite cardinal $\kappa$ there is an indecomposable (i.e., not expressible as a nontrivial direct sum) commutative group $G$ of cardinality $\kappa$. I would however be interested in hearing other examples or other takes on Question 1.
My real question is Question 2: presumably the answer is either yes or unknown, or people would mention the stronger result when Kaplansky's Theorem is discussed. A theorem of Bass that M. Reyes pointed out to me in his answer to another recent question of mine on modules is relevant in this regard: obviously an affirmative answer to Question 2 must involve an infinitely generated projective module, and if $R$ is Noetherian and connected then every infinitely generated projective module is free, hence a direct sum of singly generated submodules!