A fraction field is not finitely generated over its subdomain

Question

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 $R$-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!

Answer 1

The fraction field $K$ has the structure of an $R$-module, and for any maximal ideal $\mathfrak{m}$ of $R$, it also naturally has the structure of an $R_\mathfrak{m}$-module. If $K$ is finitely generated as an $R$-module, then $K$ must also be finitely generated as an $R_\mathfrak{m}$-module because $R_\mathfrak{m}\supseteq R$. Thus we are reduced to the case when $R$ is local.

Let $R$ be a local domain with maximal ideal $\mathfrak{m} \neq 0$. Clearly, $\mathfrak{m}K=K$. But the Jacobson radical of $R$ is just $\mathfrak{m}$, so if $K$ is finitely generated as an $R$-module, then Nakayama's lemma implies that $K=0$, which is a contradiction.

Answer 2

Zev's proof uses the existence of maximal ideals, and is therefore not constructive. But it is not hard to make the proof constructive, and actually more generally:

Let $R$ be a ring and $R'$ a localization of $R$. If $R'$ is finitely generated and torsionfree over $R$, then $R=R'$.

Proof. Let $s \in R$ such that $s^{-1} \in R'$. Then $R' = sR'$. Generalized Nakayama's Lemma implies that there is some $a \in R$ such that $(1-as)R'=0$. Since $R'$ is torsionfree, this means $1=as$. So $R'$ has no more units than $R$, i.e. $R=R'$.