Localization of a ring

In algebra, localization is a method to systematically add a new multiplicative inverse ring R elements. If you want to, that the elements of a subset S of R is invertible, then we construct a new ring after "Localization from R to S", and a ring homomorphism of R, S in units of maps. and this homomorphism satisfy the universal property of the " best choice ".

In this article we restrict ourselves to commutative rings with unit element 1 In a ring without identity is Invertierbarkeitsfragen not provide or only after adjunction of the element. For a generalization to the case of non- commutative rings see Ore condition.

• 3.1 localization at an element
• 3.2 Localization for a prime ideal
• 3.3 Total quotient ring

• 4.1 prime ideals

• 5.1 Properties

Word origin

The use of the term "localization" arises in algebraic geometry: If R is a ring of real - or complex-valued functions on a geometric object (such as an algebraic variety) and aims to analyze the behavior of the functions near a point p, then choose S for the set of functions, and which are located, according to p R localization then only contain information about the behavior of the functions of P near P not 0.

Definition

Localization of a general commutative ring

Let R be a commutative ring with 1 and S be a subset of R. Since the product of units is again a unit, 1 is a unit, and we want to make the elements of S to units, we can enlarge S and 1 and all products Add elements from S to S; so we do the same, that S is multiplicatively closed and contains the identity element. On the Cartesian product then we introduce an equivalence relation:

The occurring in the definition of equivalence relation factor t is necessary for the transitivity of the equivalence relation, if the present ring is not zero divisors. The equivalence class of a pair we write as a fraction

Addition and multiplication of equivalence classes are defined analogously to the usual fraction calculation rules ( the well- definedness, ie the independence of the choice of the Special Representative is to show ):

Use the links so defined, we get a ring. The figure

With is a (not necessarily injective ) homomorphism and independent of the choice of s

Localization of an integrity ring

In the simplest case, R is an integral domain. Here we distinguish whether S contains 0 or not.

Is, then comes for localization only the zero ring in question because he is the only ring in which 0 is the unit. So we define, if 0 is in S.

If 0 is not an element of S, so simplifying the above equivalence relation, as follows because of the reduction law in force in integrity rings: iff. Specifically is completed multiplicative, and the above structure is coincident with the known construction of the quotient body.

Localizations for a complete multiplicative subset can then be found as follows in the quotient field K of R. The partial ring of K, which consists of all apertures, the counter is R and the denominator in the S, has the desired properties: The canonical embedding of R to K is a homomorphism, which is even injective, and the elements of S are invertible. This ring is the smallest part of ring C, which contains R and S in which the elements are inverted.

Here are some examples of localizations of Z with respect to different subsets of S:

• Localized to Z with respect to the set of odd integers, one obtains the ring of all rational numbers with odd denominators. The use of the " (2)" will be explained below.
• Localized to Z with respect to the set of even numbers without the leading 0, you get all Q, because every rational number can be added via expansion with 2 can be represented as a fraction with denominator straight.
• Localized to Z based on the amount of powers of two, we obtain the ring of dual fraction. These are precisely the rational numbers whose binary representation has only a finite number of decimal places.

Category Theoretical Definition

The localization of a ring by a subset can be theoretically defined as follows category:

If R is a ring and a subset, so the set of all algebras which are such that under the canonical injection of each element is mapped to a unit, a category of algebras homomorphisms as morphisms -. The localization of after then is the initial object of this category.

This corresponds to the above-given algebraically more accessible definition as it is commonly found in textbooks on commutative algebra.

Universal property

The " best choice " of the ring and the homomorphism is defined by the fulfillment of a universal property:

This corresponds to the category theoretical definition as the initial object. The above algebraic structure is a ring for which one can prove this universal property.

Common types of localization

Isolation of an element

By setting, can you all powers of an element to the denominator. Common spellings for this are, or. The localization obtained is canonically isomorphic to, where the isomorphisms fix pointwise and mapping (or vice versa).

Localization for a prime ideal

If a prime ideal referred to, it is called for by the " localization " or "after". The resulting ring is local with the maximal ideal. Is precisely the above mentioned homomorphism, so is a for inclusion preserving bijection. The above ring is a prime number is an example of this construction.

There is no zero divisors, one can form the quotient field. It then applies.

It is the localization for a prime ideal also interpreted as follows: Taking the whole to elements of as functions on the spectrum of whose value at a point the image in the residue field is so " is " the local ring at out rupture, resulting in denominator functions that do not vanish at, " ie through which you can share with locally ".

"All Finished " is a local property, ie for a zero-divisor- free ring are equivalent:

• Is completely finished
• Is integrally closed for all prime ideals
• Is integrally closed for all maximal ideals

Total quotient ring

The total quotient ring of a ring is the localization of the amount of non- zero divisor of. It is the "strongest " localization, for which the localization Figure

Is injective. Is an integral domain, so is the total quotient ring of the quotient field of.

Ideal theory of localization

It was completed a commutative ring R and multiplicative. Denote the canonical homomorphism.

Then for any ideal

In particular, therefore, any ideal of the image of an ideal of R.

Prime ideals

The figure

Is bijective. The prime ideals of localization are therefore just the images ( under f) of prime ideals of R, which have in common with the set S is not an element.

The location for a prime ideal thus supplies a ring that has only one maximal ideal ( the image of ). Thus, the ring is a local ring with maximal ideal, which justifies the name localization. Prime ideals, it may, however, give the localization more, about the localization of a integral domain, which itself is also an integral domain, the zero ideal. Other prime ideals can be excluded when R is at most one-dimensional or, in particular a Dedekind area.

Localization of modules

R is a commutative ring having 1, S is a multiplicative subset of R and M is a RF module, the localization of M with respect to S is defined as the amount of S -1M of the equivalence class of pairs ( M, S), also written m / s, where two couples to be equivalent if there exists an element s of S such that

Applies. S -1M -1R is an S module.

According to the case of rings, you also writes Mr or MP for elements r and maximal ideals P of R.

The localization of a module also has a universal property: every R- homomorphism of M into a module N in which all elements by the elements of S " divisible ", ie the left multiplication by an element of S is a module isomorphism, can be continued uniquely to an R - homomorphism S -1M → N. This means that one can describe the location of a module as a tensor product of:

Properties

Let R be a commutative ring and M, N two R-modules, and complete multiplicative. Then we have

• For R- submodules applies. The statement is false for infinite cuts in general.
• Also for localization of a finitely generated module, there is a criterion when the delocalization provides the zero module:

Thus, the location is exactly zero when an element that cancels all modules is included in the set S. In the case of a module infinitely generated this criterion is no longer valid.