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In [[abstract algebra]], '''localization''' is a systematic method of adding multiplicative inverses to a [[ring (mathematics)|ring]]. Given a ring ''R'' and a subset ''S'', one wants to construct some ring ''R*'' and [[ring homomorphism]] from ''R'' to ''R*'', such that the image of ''S'' consists of ''[[Unit (ring theory)|units]]'' (invertible elements) in ''R*''. Further one wants ''R*'' to be the 'best possible' or 'most general' way to do this - in the usual fashion this should be expressed by a [[universal property]]. The localization of ''R'' by ''S'' is also denoted by ''S''<sup> -1</sup>''R''.
The term ''localization'' originates in [[algebraic geometry]]: if ''R'' is a ring of [[function (mathematics)|function]]s defined on some geometric object ([[algebraic variety]]) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions which are non-zero at ''p'' and localizes ''R'' with respect to ''S''. The resulting ring ''R*'' contains only information about the behavior of ''V'' near ''p''. Cf. the example given at [[local ring]].
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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 ''microlocalization'', connecting with numerous other branches. The ''micro-'' tag is to do with connections with Fourier theory, in particular.
== See also
* [[Localization of a module]] * [[localization of a category]]. ==External links==
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