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  "body": "\n## Summary\n\nOre localization builds a ring of fractions $R S^{-1}$ of a noncommutative [[ring|0B1261B5ED]] $R$.\nIn it the elements of a chosen set $S$ become units, and every element has the form $a s^{-1}$.\nIt exists when $S$ satisfies the [[Ore condition|TJP4ARC2X7]] together with a reversibility condition.\nIt extends the commutative [[localization|4C1AE84N5N]] and is the main tool to build division rings of fractions of noncommutative domains and [[classical rings of quotients|C5971PGKPH]].\n\n## Explanations\n\n### L1 human\n\nFrom the integers one builds the rationals by allowing division by nonzero integers.\nOre localization does the same for a noncommutative ring, but only on one side: every new element is a fraction $a s^{-1}$ with the denominator on the right.\nFor this to give a ring, one must be able to rewrite $s^{-1} a$ as some $b t^{-1}$, and this is what the Ore condition guarantees.\nFor example, the polynomial ring $k[x]$ over a field localizes to the field $k(x)$ of rational functions, and the [[Weyl algebra|98QE7KY55W]] localizes to a division ring.\n",
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