It's well known that if $K$ is a finitely generated extension of some field $E$, then any intermediate field $F$, $E\subseteq F\subseteq K$, is also finitely generated over $E$.
I'm curious, does the same hold for rings?
Say $S$ is a ring, and $R\supset S$ is a finitely generated extension of $S$. If $T$ is any intermediate ring, is it necessarily true that $T$ is finitely generated over $S$ as a ring?
Is it as simple as saying that for any $t\in T$, $T$ can be generated by the generators of $R$ over $S$? I feel unsure about this statement, since it's not clear to me that the generators of $R$ over $S$ need be in $T$.
If not, what is an example what shows otherwise? Thanks.
Best Answer
Let $k$ be a field. Then $k[x, y]$ is a finitely generated extension of $k$, yet the subring $R\subseteq k[x,y]$ generated by $\{xy^i:i\geq0\}$ is not finitely generated.
Life would be very much simpler in some respects if the answer were yes :)