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:
“I’m a very smart guy. I haven’t a feeling or a scruple in the world. All I have the itch for is money. I am so money greedy that for twenty-five bucks a day and expenses, mostly gasoline and whisky, I do my thinking myself, what there is of it; I risk my whole future, the hatred of the cops ... I dodge bullets and eat saps, and say thank you very much, if you have any more trouble, I hope you’ll think of me, I’ll just leave one of my cards in case anything comes up.”
—Raymond Chandler (1888–1959)