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  "body": "\n## Summary\n\nThe localization $S^{-1}R$ of a [[commutative ring|0310REJJY8]] $R$ at a [[multiplicative set|AN9SKEZ3VY]] $S$ is the ring of fractions $a/s$ with $a \\in R$ and $s \\in S$.\nIt is the most general ring in which the elements of $S$ become [[units|0XVEZDE9J8]].\nIt generalizes the construction of the [[field of fractions|0F2PTGX1A2]] of an [[integral domain|0QVD7WBFGN]].\nLocalization lets one study a ring \"near\" a [[prime ideal|00MHHZ9SNE]], which is the algebraic form of looking at a space near a point.\n\n## Explanations\n\n### L1 human\n\nLocalization means: allow some new denominators.\nStarting from the integers and allowing every nonzero denominator gives the rational numbers.\nAllowing only the powers of $2$ gives the fractions $a / 2^k$, such as $3/8$, but not $1/3$.\nIn general one picks a set $S$ of allowed denominators that contains $1$ and is closed under products.\nThe [[definition|KET2GSPZFG]] explains how to compare two fractions when the ring has zero divisors.\n",
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