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  "body": "\n## Summary\n\nA ring extension $A \\subseteq B$ is integral if every element of $B$ is [[integral|1EHN4KPSFS]] over $A$.\nIt is the ring analogue of an [[algebraic extension|0PTBM7K0R2]] of fields.\nIntegral extensions keep the prime ideals close to those of the base.\nThe lying over, going-up and incomparability theorems hold, and the two rings have the same Krull dimension.\nEvery [[finite|1NBACAEC2V]] ring extension is integral.\nIn geometry an integral extension gives a closed map of spectra whose fibres have dimension zero.\n\n## Explanations\n\n### L1 human\n\nAn integral extension adds to a ring only elements that satisfy monic equations over it.\nThe Gaussian integers $\\Z[i]$ are an integral extension of $\\Z$, since every $a + bi$ is a root of $x^2 - 2a x + (a^2 + b^2)$.\nThe rational numbers are not an integral extension of $\\Z$, since $1/2$ is not integral.\nA picture: the parabola $y^2 = x$ projected to the $x$-axis corresponds to $k[x] \\subseteq k[x, y]/(y^2 - x)$, an integral extension, and every point of the axis has one or two points above it.\n",
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