Localization of a ring

From Academic Kids

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.

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.

Commutative case

Since the product of units is a unit and since ring homomorphisms respect products, we may and will assume that S is multiplicatively closed, i.e. that for s and t in S, we also have st in S. For the same reason, we also assume that 1 is in S.

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 the r/s 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. See for example dyadic fraction, for the case R the integers and S the powers of 2.

For general commutative rings, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of "fractions" with denominators coming from S; in contrast with the integral domain case, one can now safely 'cancel' from numerator and denominator only elements of S.

This construction proceeds as follows: on R × S define an equivalence relation ~ by setting (r1,s1) ~ (r2,s2) iff there exists t in S such that

t(r1s2r2s1) = 0.

We think of the equivalence class of (r,s) as the "fraction" r/s, and using this intuition, the set of equivalence classes R* can be turned into a ring; the map j : RR* which maps r to the equivalence class of (r,1) is then a ring homomorphism.

The above mentioned universal property is the following: the ring homomorphism j : RR* maps every element of S to a unit in R*, and if f : RT is some other ring homomorphism which maps every element of S to a unit in T, then there exists a unique ring homomorphism g : R*T such that f = g o j.

For example the ring Z/nZ where n is composite is not an 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 ab with a and b coprime and greater than 1, then Z/nZ is by the Chinese remainder theorem isomorphic to Z/aZ × Z/bZ. If we take S to consist only of (1,0) and 1 = (1,1), then the corresponding localization is Z/aZ.

Two classes of localizations occur commonly in commutative algebra and algebraic geometry:

  • The set S consists of all powers of a given element r. In the theory of the spectrum of a ring, these localizations are used to identify basic open sets in Spec(R).
  • S is the complement of a given prime ideal P in R (this is a multiplicatively closed set). In this case, one also speaks of the "localization at P".

Some properties of the localization R* = S -1R:

  • S-1R = {0} if and only if S contains 0.
  • The ring homomorphism RS -1R is injective if and only if S does not contain any zero divisors.
  • There is a bijection between the set of prime ideals of S-1R and the set of prime ideals of R which do not intersect S. This bijection is induced by the given homomorphism RS -1R.
  • In particular: after localization at a prime ideal P, one obtains a local ring.

Non-commutative case

Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists is the Ore condition.

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.

See also: Localization of a module, localization of a category.

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