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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

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Bass showed, in Corollary 4.5 of

Bass, H., Big projective modules are free, Ill. J. Math. 7, 24-31 (1963). ZBL0115.26003,

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 $J\subseteq I$.

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