Subrings of polynomial rings over the complex plane

Question

I have the following questions:

(i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian?

(ii) Are there Noetherian rings containing the complex field which are not f.g. as a ring over the complex field (maybe the field of rational functions over the complex plane? is this f.g. as a ring over the complex field?)?

Answer 1

(i) No. Take $\mathbb C[x,xy,xy^2,\dots]$.

(ii) Yes. Your example is ok: $\mathbb C(x)$ is not finitely generated over $\mathbb C$, otherwise we apply the Zariski's Lemma and get that the field extension $\mathbb C\subset\mathbb C(x)$ is finite and this is false.