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  "body": "\n## Summary\n\nLet $M$ be a [[module|0NZ8JA25WK]] over a [[commutative ring|0310REJJY8]] $R$, and let $S$ be a [[multiplicative set|AN9SKEZ3VY]].\nThe localization of $M$ at $S$ is the module $S^{-1}M$ of fractions $m/s$ with $m \\in M$ and $s \\in S$.\nIt is a module over the [[localization|4C1AE84N5N]] $S^{-1}R$.\nLocalization of modules preserves [[exact sequences|JB7ZBFBA9C]], so it is a basic tool to reduce statements about modules to the local case.\n\n## Explanations\n\n### L1 human\n\nA module is like a vector space whose scalars come from a ring.\nLocalizing the module means allowing the same new denominators as in the ring.\nElements killed by some allowed denominator become zero.\nFor example, over $\\Z$ the module $\\Z/2\\Z$ disappears once you may divide by $2$, because $1 = 2 \\cdot (1/2)$ and $2 \\cdot 1 = 0$ there.\nThe [[definition|S357810Z25]] gives the construction.\n",
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