Cohn localization

Remarks

$R_\Sigma$ exists for every $R$ and $\Sigma$, and it is unique up to a unique isomorphism over $R$. It 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 $A \bar A = \bar A A = I_n$. The map $\lambda$ need not be injective, and $R_\Sigma$ may be the zero ring. Example: let $R$ be commutative and let $\Sigma$ be the set of $1 \times 1$ matrices $(s)$ with $s$ in a multiplicative set $S$. A homomorphism is $\Sigma$-inverting exactly when it sends each $s \in S$ to a unit, so $R_\Sigma$ is the localization $S^{-1}R$. More 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 $R S^{-1}$. Non-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 $\Z[1/2]$. For the free algebra $k\langle X \rangle$ over a field $k$ and $\Sigma$ the set of full matrices, $R_\Sigma$ is a division ring, the free field.