Localization of A Ring - Non-commutative Case

Non-commutative Case

Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists is the Ore condition.

One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.