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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 to R* to be the 'best possible' way to do this - in the usual fashion this should be expressed by a universal property.

The significant cases can be isolated: we might as well take S to be closed under multiplication (this is no loss of generality); and if 0 lies in S then we can take R* = 0, since if 0 is invertible in a ring it is the only element.

In case R is an integral domain there is an easy construction given the field of fractions K of R: we take R* to be the subring of K of the r/s with s in S. In this case the homomorphism from R to R* is injective: but that will not be the case in general.

For example Z/nZ where n is composite is not a integral domain. When n is a prime power it is a finite local ring, and its elements are either units or nilpotent. This implies it can be localized only to a zero ring. But when n can be factorised as a'b with a and b coprime and greater than 1, Z/nZ is by the Chinese remainder theorem Z/aZ x Z/bZ, up to isomorphism. We can take S to consist only of (1,0) and 1 = (1,1). Then the corresponding localization is Z/aZ.

In fact localization, while making sense for any ring, is a basic tool for commutative rings such as these. In the theory of the spectrum of a ring it is used to identify basic open sets in Spec(R) in terms of sets S that are the powers of some given r in R. The corresponding localized ring is then easily constructed as R* = R[X]/(rX - 1). The image of X has been 'forced' to be the inverse of r.

In the general case (for commutative rings), it is a formal matter to imitate this with enough variables in a polynomial ring, and to check the universal property. What is interesting is then to check that there is a fraction description, elements being rs-1 = r/s. In contrast with the integral domain case, one can now safely 'cancel' from numerator and denominator only elements of S.

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.

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