Editing Localization (commutative algebra) (section)

== Non-commutative case ==
Localizing [[non-commutative ring]]s is more difficult. While the localization exists for every set ''S'' of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved 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''<sup>&minus;1</sup> for a differentiation operator ''D''. This is done in many contexts in methods for [[differential equation]]s. There is now a large mathematical theory about it, named [[microlocal analysis|microlocalization]], connecting with numerous other branches. The ''micro-'' tag is to do with connections with [[Fourier theory]], in particular.