Normal ring

Remarks

Example: $\Z \times \Z$ is normal. Its prime ideals are $\mathfrak{q} \times \Z$ and $\Z \times \mathfrak{q}$ for a prime ideal $\mathfrak{q}$ of $\Z$, and the localizations are $\Z_{\mathfrak{q}}$, which are integrally closed domains. It is not a domain, since $(1, 0) \cdot (0, 1) = 0$.

Non-example: $R = k[x, y]/(y^2 - x^3)$ over a field $k$ is not normal. At the prime $\mathfrak{p} = (x, y)$, the element $y/x$ of the fraction field of $R_{\mathfrak{p}}$ is a root of $z^2 - x$, but it is not in $R_{\mathfrak{p}}$.

An integral domain is normal if and only if it is integrally closed. A Noetherian ring is normal if and only if it is a finite product of integrally closed domains. Some authors use the word "normal ring" only for integrally closed domains.