---
tag: G449HX8H25
kind: concept
name: Cohn localization
msc: [16S85, 16S10]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
    locator: Non-commutative case
---

## Summary

Cohn localization, also called universal localization, inverts a chosen set $\Sigma$ of square matrices over a [[ring|0B1261B5ED]] $R$ 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|SG7AQ9T8HD]], which needs the Ore condition.
P. M. Cohn used it to construct the free field, the universal field of fractions of a free algebra.

## Explanations

### L1 human

For commutative rings, one makes fractions by declaring some elements invertible.
In a noncommutative ring, inverting single elements is not always enough to get a good ring of fractions.
Cohn's idea is to declare whole square matrices invertible, since a field of fractions must make many matrices invertible at once.
The universal localization is the "freest" ring in which the chosen matrices have inverses.
For a set of $1 \times 1$ matrices in a commutative ring, it is the usual [[localization|4C1AE84N5N]].