Is a regular ring a domain

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.

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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).