symmetric monoidal (∞,1)-category of spectra
Let be a mulitplicative family of matrices over a (typically noncommutative) ring .
We say that a map of rings is -inverting if all matrices over where are invertible in . The Cohn localization of a ring , is a homomorphism of rings which is initial in the category of all -inverting maps (which is the subcategory of coslice category ).
Given a ring and a family of morphisms in the category Mod of (say left) finitely generated projective -modules, we say that a morphism of rings is -inverting if the extension of scalars from to along
T \otimes_R (-) \colon R Mod \to T Mod
sends all morphisms of into isomorphism in the category of left -modules.
P. M. Cohn has shown that there is a universal object in the category of -inverting morphisms. The ring (and more precisely the universal morphism itself) are called the universal localization or Cohn localization of the ring at .
Cohn localization induces a hereditary torsion theory, i.e. a localization endofunctor on the category of all modules, but it lacks good flatness properties at the level of full module category. However when restricted to the subcategory of finite-dimensional projectives it has all good properties – it is not any worse than Ore localization.
Universal localization is much used in algebraic K-theory, algebraic L-theory and surgery theory – see Andrew Ranicki’s slides in the references at Cohn localization and his papers, specially the series with Amnon Neeman.
One can also look at localization with inverses just from one side: