We know that for general modules over a commutative ring with $1$, you can't always extract a basis from a generating set.
This makes me think that maybe there should be free modules of infinite rank which could be finitely generated. Do such things exist?
Best Answer
Suppose we had such a horrible thing, a surjection: $$R^{\oplus n} \twoheadrightarrow R^{\oplus I}$$where $I$ is something infinite (or just finite and $> n$). Pick any maximal ideal of $R$ and tensor up with $R/\mathfrak{m}$, it's right exact so we still have: $$R^{\oplus n} \otimes R/\mathfrak{m} \twoheadrightarrow R^{\oplus I} \otimes R/\mathfrak{m}$$But these are isomorphic to: $$R/\mathfrak{m}^{\oplus n} \twoheadrightarrow R/\mathfrak{m}^{\oplus I}$$a surjection of vector spaces.
So, can't happen!