Encyclopedia > Localization of a ring

  Article Content

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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Kings Park, New York

... or Latino of any race. There are 5,480 households out of which 36.4% have children under the age of 18 living with them, 65.1% are married couples living together, ...

This page was created in 24.4 ms