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Let $R$ be a (not necessarily commutative) ring.

If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that

$M \subseteq P' \subseteq P$ ?

Kaplansky's "Projective modules" (Ann. of Math. Vol. 68, No. 2, pp. 372-377), Section 5, Lemma 3, answers this affirmatively in the situation $R$ commutative semi-hereditary. A few more positive results are known, e.g. work of Bass, but also negative ones and it appears in general one should expect this to fail.

Nonetheless, I wonder is there a clear criterion / philosophy / guideline what to expect for a given ring, e.g., are there sufficient criteria for the answer being negative?

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