I'm looking for proofs of the following fact.
Suppose that $R$ is a domain which is not a field with fraction field $K$. Then $K$ is not finitely generated as an $R$-module.
I know this fact is true, at least, when $R$ is Noetherian and I guess it is true in general. I know two proofs, one when $R$ is Noetherian and one (very indirect) when $R$ is Noetherian local. Do you know any direct proof for any arbitrary domain $R$? Thanks!