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  "body": "\n## Summary\n\nCohn localization, also called universal localization, inverts a chosen set $\\Sigma$ of square matrices over a [[ring|0B1261B5ED]] $R$ in the most general way.\nIt produces a ring $R_\\Sigma$ and a ring homomorphism $R \\to R_\\Sigma$ over which every matrix in $\\Sigma$ becomes invertible, universal among such homomorphisms.\nIt exists for every ring and every set of matrices, unlike [[Ore localization|SG7AQ9T8HD]], which needs the Ore condition.\nP. M. Cohn used it to construct the free field, the universal field of fractions of a free algebra.\n\n## Explanations\n\n### L1 human\n\nFor commutative rings, one makes fractions by declaring some elements invertible.\nIn a noncommutative ring, inverting single elements is not always enough to get a good ring of fractions.\nCohn's idea is to declare whole square matrices invertible, since a field of fractions must make many matrices invertible at once.\nThe universal localization is the \"freest\" ring in which the chosen matrices have inverses.\nFor a set of $1 \\times 1$ matrices in a commutative ring, it is the usual [[localization|4C1AE84N5N]].\n",
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