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 $k[{x}_{1},{x}_{2},\dots ]$, 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 $f,g\in k[{x}_{1},{x}_{2},\dots ]$, then there exists $M\in \mathbb{N}$ such that $f,g\in k[{x}_{1},\dots ,{x}_{M}]$, so if we think of $fg=0$ in $k[{x}_{1},\dots ,{x}_{M}]$, we get that $f=0$ or $g=0$, and we are done.