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Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand?

N.B.: I asked this alreadyalready on math.stackexchange.com without much luck.

Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand?

N.B.: I asked this already on math.stackexchange.com without much luck.

Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand?

N.B.: I asked this already on math.stackexchange.com without much luck.

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Lemma on infinitely generated projective modules

Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand?

N.B.: I asked this already on math.stackexchange.com without much luck.