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  "body": "\n## Definition\n\nA [[scheme|5A35TYZD02]] $X$ is *integral* if $X$ is nonempty and for every nonempty open set $U \\subseteq X$ the ring $\\mathcal{O}_X(U)$ is an [[integral domain|0QVD7WBFGN]].\n\n## Remarks\n\nA scheme is integral if and only if it is [[reduced|ZXSN8D4MCB]] and its underlying space is irreducible.\nAn affine scheme $\\operatorname{Spec} A$ is integral if and only if $A$ is an integral domain.\nThen its [[generic point|3A19C4YXGE]] is $(0)$, and the local ring there is the [[field of fractions|0F2PTGX1A2]] of $A$, the function field.\nExample: $\\operatorname{Spec} \\Z$ is integral, since $\\Z$ is a domain and each $\\mathcal{O}(D(n)) = \\Z[1/n]$ is a domain.\nNon-example: $\\operatorname{Spec} k[x, y]/(xy)$ for a field $k$ is reduced but not integral, since $x y = 0$ with $x \\ne 0$ and $y \\ne 0$.\nThe open set $U = \\emptyset$ is excluded because $\\mathcal{O}_X(\\emptyset)$ is the zero ring, which is not a domain.\n",
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