Localization of a ring

In abstract algebra, localization is a systematic method of adding multiplicative inverses to a 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 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 -1R.

Related Topics:
Abstract algebra - Ring - Ring homomorphism - Universal property

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The term localization originates in algebraic geometry: if R is a ring of functions 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.

Related Topics:
Algebraic geometry - Function - Algebraic variety - Local ring

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