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](http://alpha.math.uga.edu/%7Epete/integral.pdf). In particular, it raises two natural questions:



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> Question 1: Is there a ring $R$ and an $R$-module $M$ which is *not* a direct sum of countably generated submodules?
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> Question 2: Is there a ring $R$ and a projective $R$-module $P$ which is *not* a direct sum of finitely generated submodules?
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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!



Example of a projective module which is not a direct sum of f.g. submodules?
For question two the example that is given most frequently seems to be that of the ring $R$ of continuous real valued functions on $[0,1]$ and the ideal of all functions $f$ which vanish on some interval $[0,\epsilon(f)]$ where $\epsilon(f)\in (0,1)$. This ideal is countably generated and projective but not a direct sum of finitely generated submodules. You might also want to take a look at the article "When every projective module is a direct sum of finitely generated modules" by W. McGovern, G. Puninski and P. Rothmaler.


