Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or at least , is every finitely generated submodule of $P$ contained in some proper direct summand of $P$ ? 


This question has been motivated by [Lemma on infinitely generated projective modules](https://mathoverflow.net/questions/72788/lemma-on-infinitely-generated-projective-modules)



Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ?
Bass showed, in Corollary 4.5 of


*Bass, H.*, Big projective modules are free, Ill. J. Math. 7, 24-31 (1963). [ZBL0115.26003](https://zbmath.org/?q=an:0115.26003),


that every projective module for a connected Noetherian commutative ring is either finitely generated or free (and hence a direct sum of finitely generated modules).


Every Noetherian commutative ring is a finite direct product of connected rings (otherwise it is easy to construct an infinite ascending chain of ideals). So every projective module $P$ for a Noetherian commutative ring is a direct sum $\bigoplus\_{i\in I}P\_i$ of finitely generated modules, and so every finitely generated submodule lies in $\bigoplus\_{j\in J}P\_j$ for some finite subset $J\subseteq I$.


