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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nThe ring $R$ is *normal* if for every [[prime ideal|00MHHZ9SNE]] $\\mathfrak{p}$ of $R$, the [[localization|X6JD4AG973]] $R_{\\mathfrak{p}}$ is an [[integrally closed domain|H20P600HQS]].\n\n## Remarks\n\nExample: $\\Z \\times \\Z$ is normal.\nIts 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.\nIt is not a domain, since $(1, 0) \\cdot (0, 1) = 0$.\n\nNon-example: $R = k[x, y]/(y^2 - x^3)$ over a field $k$ is not normal.\nAt 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 $R_{\\mathfrak{p}}$.\n\nAn integral domain is normal if and only if it is integrally closed.\nA Noetherian ring is normal if and only if it is a finite product of integrally closed domains.\nSome authors use the word \"normal ring\" only for integrally closed domains.\n",
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