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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $S \\subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]], and let $M$ be an $R$-[[module|0NZ8JA25WK]].\nOn $M \\times S$, let $(m, s) \\sim (n, t)$ if there is $u \\in S$ with $u(tm - sn) = 0$.\nThis is an [[equivalence relation|0G8VEZ3WTK]]; write $m/s$ for the class of $(m, s)$.\nThe *localization* of $M$ at $S$ is the set $S^{-1}M$ of these classes with the operations\n$$\\frac{m}{s} + \\frac{n}{t} = \\frac{tm + sn}{st}, \\qquad \\frac{a}{t} \\cdot \\frac{m}{s} = \\frac{am}{ts}$$\nfor $a \\in R$.\nThese operations are well defined and make $S^{-1}M$ a module over the [[localization|4C1AE84N5N]] $S^{-1}R$.\nFor a [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$ of $R$, one writes $M_{\\mathfrak p}$ for $S^{-1}M$ with $S = R \\setminus \\mathfrak p$.\n\n## Remarks\n\nThere is a natural isomorphism of $S^{-1}R$-modules $S^{-1}R \\otimes_R M \\cong S^{-1}M$, given by $(a/s) \\otimes m \\mapsto am/s$; see [[tensor product of modules|F0HQXQ0HCD]].\nThe functor $M \\mapsto S^{-1}M$ is exact, so $S^{-1}R$ is a [[flat|73RYYEVDKA]] $R$-module.\n\nExample: let $R = \\Z$, $M = \\Z/3\\Z$ and $\\mathfrak p = 2\\Z$.\nThen $3 \\in S = \\Z \\setminus 2\\Z$, and $m/s = 3m/(3s) = 0$ for every $m$, so $M_{(2)} = 0$.\nFor $\\mathfrak p = 3\\Z$, the elements of $S$ act invertibly on $\\Z/3\\Z$, and $M_{(3)} \\cong \\Z/3\\Z$.\n\nNon-example: the relation \"$tm = sn$\" without the factor $u$ is in general not transitive, as for the [[localization of a ring|4C1AE84N5N]], which is the case $M = R$.\n",
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