Let $R$ be a commutative $k$-algebra, where $k$ is a field of characteristic zero.
Could one please give an example of such $R$ which is also:
(i) Not affine (= infinitely generated as a $k$-algebra).
and
(ii) Not an integral domain (= has zero divisors).
My first thought was $k[x_1,x_2,\ldots]$, the polynomial ring over $k$ in infinitely many variables, but unfortunately, it satisfies condition (i) only. It is not difficult to see that it is an integral domain: If $fg=0$ for some $f,g \in k[x_1,x_2,\ldots]$, then there exists $M \in \mathbb{N}$ such that $f,g \in k[x_1,\ldots,x_M]$, so if we think of $fg=0$ in $k[x_1,\ldots,x_M]$, we get that $f=0$ or $g=0$, and we are done.
Thank you very much!
For instance, $k[X_n\,:\, n\in\Bbb N]/(X_n^2\,:\, n\in\Bbb N)$: the ring of polynomials in infinitely many variables quotiented by the ideal generated by the square of the variables.