Normal ring which is not integrally closed integral domain

Question

Here, normal ring is a integral domain $A$ whose localization at every prime ideal is integrally closed integral domain.

Then, it is known that integrally closed integral domain is normal ring.
But I heard the converse does not hold in general.

Could you give me an example of the titled ring?

Thank you in advance.

Answer 1

To me, a normal ring means a ring whose localizations at prime ideals are domains which are integrally closed in their field of fractions.

$\mathbb Z/6\mathbb Z$ is such a ring, since its localizations are fields.