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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $\\Sigma$ be a set of square [[matrices|0VD9ZSWWJG]] over $R$, of possibly different sizes.\nA [[ring homomorphism|0VTH8KPCG2]] $f \\colon R \\to S$ is *$\\Sigma$-inverting* if for every $A \\in \\Sigma$ the matrix $f(A)$, obtained by applying $f$ to each entry, is invertible over $S$.\nThe *Cohn localization* (or *universal localization*) of $R$ at $\\Sigma$ is a ring $R_\\Sigma$ with a $\\Sigma$-inverting homomorphism $\\lambda \\colon R \\to R_\\Sigma$ with this property.\nEvery $\\Sigma$-inverting homomorphism $f \\colon R \\to S$ factors as $f = g \\circ \\lambda$ for exactly one ring homomorphism $g \\colon R_\\Sigma \\to S$.\n\n## Remarks\n\n$R_\\Sigma$ exists for every $R$ and $\\Sigma$, and it is unique up to a unique isomorphism over $R$.\nIt can be built by adjoining, for each $n \\times n$ matrix $A \\in \\Sigma$, $n^2$ new generators forming a matrix $\\bar A$, with the relations $A \\bar A = \\bar A A = I_n$.\nThe map $\\lambda$ need not be injective, and $R_\\Sigma$ may be the zero ring.\nExample: let $R$ be [[commutative|0310REJJY8]] and let $\\Sigma$ be the set of $1 \\times 1$ matrices $(s)$ with $s$ in a multiplicative set $S$.\nA homomorphism is $\\Sigma$-inverting exactly when it sends each $s \\in S$ to a unit, so $R_\\Sigma$ is the [[localization|4C1AE84N5N]] $S^{-1}R$.\nMore generally, for a right denominator set $S$, that is, a multiplicative set satisfying the right Ore condition and right reversibility, $R_\\Sigma$ is the [[Ore localization|SG7AQ9T8HD]] $R S^{-1}$.\nNon-example: the identity map $\\Z \\to \\Z$ is not $\\Sigma$-inverting for $\\Sigma = \\set{(2)}$, since $2$ is not a unit in $\\Z$; the localization in this case is $\\Z[1/2]$.\nFor the [[free algebra|2P3Y2FRCSZ]] $k\\langle X \\rangle$ over a field $k$ and $\\Sigma$ the set of full matrices, $R_\\Sigma$ is a division ring, the free field.\n",
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