For a (commutative, say) ring $R$, and an $R$-module $M$ it is known that $M$ is both:

- a filtered colimit of finitely generated $R$-
*sub*modules (by considering all finite subsets of $M$ and considering the submodules generated by those) - a filtered colimit of finitely presented $R$-modules (by taking finitely generated submodules, and taking more and more (but always finitely many) relations). These finitely presented modules will not in general be
*sub*modules of $M$.

**Question:** For what rings $R$ can we present any $R$-module $M$ as a filtered colimit of finitely presented submodules (with transition maps also being injective)?

For example, this is true for Noetherian rings, since then finitely generated and finitely presented is the same thing.

Equivalently, is there a condition on $R$ guaranteeing such that for any $N \subset M$, $N$ being finitely generated, there is an intermediate module $N'$, $N \subset N' \subset M$, that is finitely presented? If there is such a statement only for selected modules $M$, I'm also interested in such partial results.