Example of a normal ring

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In 'Commutative Ring Theory' by Matsumura the definition of normal ring is as follows:

A ring $R$ is called normal if for every prime ideal $\mathfrak p\subset R$, $R_{\mathfrak p}$ is an integrally closed domain.

I know that a domain is integrally closed if and only if localisation at every prime ideal gives an integrally closed domain, i.e., it is normal.

I want to have an example of a 'non-domain' which is normal. Also is there any equivalent criteria for a ring (not necessarily domain) to be a normal ring like in the domain case?

Thank you in advance.

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2 Answers

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The product of normal rings is normal and never a domain.

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For example $F\times F$ for a field $F$.

Or any of these at DaRT.

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