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→major edit: the set S in the dyadic fractions example is the complement of the set of powers of 2. |
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=== Construction ===
In case ''R'' is an [[integral ___domain]] there is an easy construction of the localization. Since the only ring in which 0 is a unit is the [[trivial ring]] {0}, the localization ''R*'' is {0} if 0 is in ''S''. Otherwise, the [[field of fractions]] ''K'' of ''R'' can be used: we take ''R*'' to be the subring of ''K'' consisting of the elements of the form <sup>''r''</sup>⁄<sub>''s''</sub> with ''r'' in ''R'' and ''s'' in ''S''. In this case the homomorphism from ''R'' to ''R*'' is the standard embedding and is injective: but that will not be the case in general. For example, the
For general [[commutative ring]]s, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of "fractions" with [[denominator]]s coming from ''S''; in contrast with the integral ___domain case, one can safely 'cancel' from [[numerator]] and [[denominator]] only elements of ''S''.
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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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