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  "body": "\n## Summary\n\nThe classical ring of quotients of a ring $R$ is obtained by inverting all regular elements of $R$.\nThese are the elements that are not [[zero divisors|0NNRDW10RZ]] on either side.\nIt is the [[Ore localization|SG7AQ9T8HD]] of $R$ at the set of regular elements, when that localization exists.\nFor an integral domain it is the [[field of fractions|0F2PTGX1A2]], and for a commutative ring it is the [[total ring of fractions|K5Z67HR9ET]].\n[[Goldie's theorem|MDS8655N0Z]] says exactly when this ring is semisimple Artinian.\n\n## Explanations\n\n### L1 human\n\nThe rationals are built from the integers by allowing division by every nonzero integer.\nFor a general ring one cannot divide by an element that kills some other nonzero element, so one only tries to divide by the regular elements.\nThe classical ring of quotients is the ring of fractions whose denominators are all the regular elements, if it exists.\nFor a ring of $2 \\times 2$ matrices over $\\Z$, it is the ring of $2 \\times 2$ matrices over $\\Q$, since an integer matrix with nonzero determinant becomes invertible over $\\Q$.\n",
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