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  "body": "\n## Definition\n\nLet $B$ be a commutative ring and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.\nThe ring $B$ is *integral over $A$*, and $A \\subseteq B$ is an *integral extension*, if every element of $B$ is [[integral|1EHN4KPSFS]] over $A$.\nA [[ring homomorphism|0GGE0Z0JQ1]] $f \\colon A \\to B$ of commutative rings is *integral* if $B$ is integral over the subring $f(A)$.\n\n## Remarks\n\nExample: $\\Z \\subseteq \\Z[i]$ is integral.\nThe element $a + bi$ with $a, b \\in \\Z$ is a root of the monic polynomial $x^2 - 2a x + a^2 + b^2$ with integer coefficients.\n\nNon-example: $\\Z \\subseteq \\Q$ is not integral, because $1/2$ is not integral over $\\Z$.\n\nAn extension is integral and of finite type if and only if it is [[finite|1NBACAEC2V]].\nIntegrality is [[transitive|ZXZMQM818D]], and it is kept by quotients and localizations.\nIf $B$ is integral over $A$, then $B/J$ is integral over $A/(J \\cap A)$ for every ideal $J$ of $B$.\nAlso $S^{-1}B$ is integral over $S^{-1}A$ for every multiplicative set $S$ of $A$.\n",
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