Any module is the colimit of its finitely generated submodules.

Fact A : If $E_i$ is a directed system of $R$-modules, and $F$ is another $R$-module. Then

$$\varinjlim(F\otimes E_{i})=F\otimes(\varinjlim E_{i}).$$

Fact B : "Every module is a direct limit of its finitely generated submodules."

How does Fact A implies Fact B?

Serge Lang claims this in his Algebra book.

• possible duplicate of Tensor products commute with direct limits – rschwieb Aug 5 '14 at 13:25
• @rschwieb: I am not asking for the proof of the expression I just wrote. I am asking how does it follow from that "Any module is a direct limit of its finitely generated submodule." – Babai Aug 5 '14 at 13:27
• Yes, that's the reduction. My thought right now is to further reduce to the case of $R=k$, a field, but I'm not having so much luck managing it. – Cass Aug 5 '14 at 14:39
• Just as a note, by the way, I don't really see why any of this is necessary. The map $f:lim E_i \rightarrow E$ is injective because direct limit preserves injectivity. $f$ is also surjective because every element $e$ of $E$ can be put into a finitely generated submodule, namely $Re$. So $f$ is an isomorphism. Maybe there's some other way with this Fact, but the proof I just gave is about as straight ahead as anyone could hope. – Cass Aug 5 '14 at 14:56
• Take the trivial directed system $F_i = E$ for all $i$ over the same index set as $E_i$. Then, we have injective maps $E_i \rightarrow F_i$ for all $i$ and these are compatible with directed system maps. Take direct limit on both sides. lim is left (and right) exact, so you get an injective map $lim E_i \rightarrow lim F_i$. But $lim F_i$ is clearly $E$. – Cass Aug 5 '14 at 15:14