Q:

Subring of a finitely generated Noetherian ring need not be Noetherian?

A common example showing that a subring of a Noetherian ring is not necessarily Noetherian is to take a polynomial ring over a field $k$ in infinitely many indeterminates, $k[x_1,x_2,\dots]$. The quotient field is then obviously Noetherian, but the subring $k[x_1,x_2,\dots]$ is not since there is an infinite ascending chain of ideals which never stabilizes.

Is there an instance of a finitely generated Noetherian ring over some ground ring $R$, that has an intermediate ring which is not finitely generated over $R$, and hence not Noetherian either?

A:

To get this question off the unanswered list, I copy Mariano's example from here.
The polynomial ring $k[x,y]$ is Noetherian (by Hilbert's basis theorem), but the subring generated by $\{xy^i:i\geq0\}$ is not finitely generated over $k$.

