Are free submodules of finitely generated modules finitely generated?
It feels like this should be true because it seems weird that a finitely generated module will have an infinite linearly independent subset, but I am unable to prove it.
Assume that the ring is commutative with unity.
If $R$ is a non-trivial commutative ring and $M$ is generated by $n$ elements then $n$ does not contain a free submodule of rank greater than $n$.
To see this assume we have an injection $R^m \hookrightarrow M$ and a surjection $R^n \twoheadrightarrow M$. As $R^m$ is free, the map to $M$ factors through $R^n$ and we get an injection $R^m \hookrightarrow R^n$. As $R$ is nonzero commutative this implies $m \leq n$ (see, for example, here).