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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nAn element $c \\in R$ is *regular* if $c$ is not a [[zero divisor|0NNRDW10RZ]] on either side: $c a = 0$ or $a c = 0$ implies $a = 0$.\nLet $C$ be the set of regular elements of $R$.\nA *right classical ring of quotients* of $R$ is a ring $Q$ containing $R$ as a [[subring|0GNCRPYXYV]] such that:\n\n1. every regular element of $R$ is a [[unit|0XVEZDE9J8]] of $Q$;\n2. every element of $Q$ has the form $a c^{-1}$ with $a \\in R$ and $c \\in C$.\n\nThe *left classical ring of quotients* is defined in the same way with elements $c^{-1} a$.\nIf $R$ has a right and a left classical ring of quotients, they coincide, and this ring is the *classical ring of quotients* of $R$.\n\n## Remarks\n\nThe right classical ring of quotients is the [[Ore localization|SG7AQ9T8HD]] $R C^{-1}$.\nIt exists if and only if $C$ satisfies the right [[Ore condition|TJP4ARC2X7]]; reversibility holds automatically, since elements of $C$ are not zero divisors.\nIt is then unique up to an isomorphism that fixes $R$.\n\nExample: for a commutative integral domain $R$, the set $C$ is $R \\setminus \\set{0}$, and the classical ring of quotients is the field of fractions.\nFor $R = \\Z$ it is $\\Q$: each nonzero integer is invertible in $\\Q$, and each rational number is $a c^{-1}$ with $a, c \\in \\Z$ and $c \\ne 0$.\n\nNon-example: the free algebra $k\\langle x, y \\rangle$ over a field $k$ is a domain, so $C$ consists of all nonzero elements.\nIt has no right classical ring of quotients, because $C$ fails the right Ore condition: $x C \\cap y R = \\emptyset$.\n\nA ring $R$ with a right classical ring of quotients $Q$ is called a *right order* in $Q$.\nA right Ore domain has a right classical ring of quotients that is a division ring.\n",
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