Non-Noetherian 0-dimensional ring

Question

I want to find a ring $R$ which satisfies

$R$ is not Noether and $\operatorname{Spec}R$ is Hausdorff.

I found the latter condition is equivalent to R's Krull dimension is $0$. So, I just need to find an example of Non-noether and $0$ dimensional ring. Are there any good examples? Thank you advance.

Answer 1

If one takes $R=K[x_1,x_2,\ldots]/(x_1^2,x_2^2,\ldots)$ to be the polynomial ring over a field in infinitely many variables modulo the ideal $(x_1^2,x_2^2,\ldots)$, then $R$ is not noetherian and $\operatorname{Spec}R$ is a single point, in particular Hausforff.