Let $R$ be a commutative $k$-algebra, where $k$ is a field of characteristic zero. Assume that $R$ satisfies the following two conditions:
(i) $R$ is not affine (= $R$ is infinitely generated as a $k$-algebra).
(ii) $R$ is not an integral domain (= $R$ has zero divisors).
After receiving an easy answer to this question, I would like to generalize it to the following question:
Is it possible to characterize all such $R$'s?
If I am not wrong, all such $R$'s are of the following form: $\frac{k[x_i]_{i \in \mathbb{N}}}{I}$, for an appropriate ideal $I$, for example $I=(x_{17}^2)$. How to find all such $I$'s?
Is it true that a general such $I$ is an ideal generated by at least one reducible polynomial? (clearly, if $h=h_1h_2 \in I$, then $\bar{h_1}\bar{h_2}=\bar{0}$).
Thank you very much!
Edit: Perhaps the power series ring in infinitely many variables (which is an integral domain) quotiented by an appropriate ideal shows that I was wrong, and not all such $R$'s are quotients of polynomial rings in infinitely many variables.