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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<sup>-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.
== See also ==
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