Is a regular ring a domain

Question

A regular local ring is a domain. Is a regular ring (a ring whose localization at every prime ideal is regular) also a domain? I am unable to find/construct a proof or a counterexample. Any help would be appreciated.

Answer 1

No. E.g. choose two regular domains and take their product; this will be regular, but not a domain.

This is more or less the general case, as I will explain: In general, a Noetherian ring, all of whose localizations at its prime ideals are domains, is a finite product of domains (and of course a finite product of domains has this localization property). So a regular Noetherian ring will be a finite product of regular domains (and conversely any such product will be regular).

Geometrically, one can think of this as follows: regularity of $A$ is a local property on Spec $A$, and Spec $A\times B$ is equal to Spec $A \coprod$ Spec $B$. So locally Spec $A\times B$ looks like either Spec $A$ or Spec $B$. In particular, local properties, such as regularity (or the condition that the localization at prime ideals be a domain) can't detect global properties (like $A$ itself being a domain).