Localization of a ring

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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 can be denoted by S^-1R or R_S.

1 Terminology
2 Construction and properties for commutative rings
3 Applications
4 Non-commutative case
5 See also
6 External links

Terminology

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 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.

In number theory and algebraic topology, one refers to the behavior of a ring or space at a number n or away from n. "Away from n" means "in a ring where n is invertible" (so a Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Z}[\textstyle{\frac{1}{n}}] -algebra). For instance, for a field, "away from p" means "characteristic not equal to p". Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Z}[\textstyle{\frac{1}{2}}]

is "away from 2", but Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}_2
or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Z}
are not.

Construction and properties for commutative rings

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.

Construction

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 ^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 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(r₁s₂ − r₂s₁) = 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 with operations that look identical to those of elementary algebra: a/s+b/t=(at+bs)/st and (a/s)(b/t)=ab/st. The map j : R → R* which maps r to the equivalence class of (r,1) is then a ring homomorphism. (In general, this is not injective; if two elements of R differ by a zero divisor with an annihilator in S, their images under j are equal.) The ring R* is sometimes called the total ring of fractions.

The above mentioned universal property is the following: the ring homomorphism j : R → R* maps every element of S to a unit in R*, and if f : R → T 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.

Examples

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.

Properties

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

S^-1R = {0} if and only if S contains 0.
The ring homomorphism R → S^-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 R → S^-1R.
In particular: after localization at a prime ideal P, one obtains a local ring, or in other words, a ring with one maximal ideal, namely the ideal generated by the extension of P.

Applications

Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec(R).

The set S consists of all powers of a given element r. The localization corresponds to restriction to the Zariski open subset U_f ⊂ Spec(R) where the function r is non-zero (the sets of this form are called principal Zariski open sets). For example, if R = K[X] is the polynomial ring and r = X then the localization produces the ring of Laurent polynomials K[X, X⁻¹]. In this case, localization corresponds to the embedding U ⊂ A¹, where A¹ is the affine line and U is its Zariski open subset which is the complement of 0.

The set S is the complement of a given prime ideal P in R. The priminess of P implies that S is a multiplicatively closed set. In this case, one also speaks of the "localization at P". Localization corresponds to restriction to the complement U in Spec(R) of the irreducible Zariski closed subset V(P) defined by the prime ideal P.

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.