Let be a commutative -algebra, where is a field of characteristic zero.
Could one please give an example of such which is also:
(i) Not affine (= infinitely generated as a -algebra).
(ii) Not an integral domain (= has zero divisors).My first thought was , 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 for some , then there exists such that , so if we think of in , we get that or , and we are done.