Cohn localization

Summary

Cohn localization, also called universal localization, inverts a chosen set $\Sigma$ of square matrices over a ring $R$ in the most general way. It produces a ring $R_\Sigma$ and a ring homomorphism $R \to R_\Sigma$ over which every matrix in $\Sigma$ becomes invertible, universal among such homomorphisms. It exists for every ring and every set of matrices, unlike Ore localization, which needs the Ore condition. P. M. Cohn used it to construct the free field, the universal field of fractions of a free algebra.