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.
Read more about this topic: Localization Of A Ring
Famous quotes containing the word case:
“The doctors are all agreed that I am suffering for want of society. Was never a case like it. First, I did not know that I was suffering at all. Secondly, as an Irishman might say, I had thought it was indigestion of the society I got.”
—Henry David Thoreau (1817–1862)