Any module is the colimit of its finitely generated submodules.

Question

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.

Answer 1

I am 100% sure that Fact A and Fact B cannot be proven from another.

[So in some sense, I don't give an answer here, but rather the meta-answer that there is no answer.]