Localization of a ring
Encyclopedia
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 often denoted by S −1R, or by RI if S is the complement
of a prime ideal
I.
An important related process is completion
: one often localizes a ring, then completes.
. If R is a ring
and S is a subset, consider the set of all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. The elements of this set form the objects of a category
, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object
of this category.
: if R is a ring of function
s 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 not 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 Z[1/n]-algebra). For instance, for a field, "away from p" means "characteristic not equal to p". Z[1/2] is "away from 2", but F2 or Z are not.
of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicative set.
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. For example, the dyadic fractions are the localization of the ring of integers with respect to the set of powers of 2. In this case, R* is the dyadic fractions, R is the integers, S is the powers of 2, and the natural map from R to R* is injective.
For general commutative ring
s, 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
(The presence of t is crucial to the transitivity of ~)
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 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 ∘ j.
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).
.
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 equation
s. 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.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, localization is a systematic method of adding multiplicative inverses to a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
from R to R*, such that the image of S consists of units
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
(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
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
. The localization of R by S is often denoted by S −1R, or by RI if S is the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
I.
An important related process is completion
Completion (ring theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
: one often localizes a ring, then completes.
Formal definition
Another way to describe the localization of a ring R at a subset S is via category theoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. If R is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
and S is a subset, consider the set of all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. The elements of this set form the objects of a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
of this category.
Terminology
The term localization originates in algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
: if R is a ring of function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s defined on some geometric object (algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions which are not 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
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
.
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although 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 Z[1/n]-algebra). For instance, for a field, "away from p" means "characteristic not equal to p". Z[1/2] is "away from 2", but F2 or 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 a submonoid of the multiplicative monoidMonoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicative set.
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 fractionsField of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
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. For example, the dyadic fractions are the localization of the ring of integers with respect to the set of powers of 2. In this case, R* is the dyadic fractions, R is the integers, S is the powers of 2, and the natural map from R to R* is injective.
For general commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s, 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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
~ by setting (r1,s1) ~ (r2,s2) iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
there exists t in S such that
- t(r1s2 − r2s1) = 0.
(The presence of t is crucial to the transitivity of ~)
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
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
. (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 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 ∘ j.
Examples
- Given a commutative ring R, we can consider the multiplicative set S of non-zerodivisors (i.e. elements a of R such that multiplication by a is an injection from R into itself.) The ring S−1R is called the total quotient ringTotal quotient ringIn abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of a domain to commutative rings that may have zero divisors. The construction embeds the ring in a larger ring, giving every non-zerodivisor of the...
of R. S is the largest multiplicative set such that the canonical mapping from R to S−1R is injective. When R is an integral domain, this is none other than the fraction field of R. - The ring Z/nZModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
where n is compositeComposite numberA composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
is not an integral domain. When n is a primePrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
power it is a finite local ringLocal ringIn abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
, and its elements are either units or nilpotentNilpotentIn mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
. This implies it can be localized only to a zero ring. But when n can be factorised as ab with a and b coprimeCoprimeIn number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
and greater than 1, then Z/nZ is by the Chinese remainder theoremChinese remainder theoremThe Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
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. - Let R = Z, and p a prime number. If S = Z - pZ, then R* is the localization of the integers at p. See Lang's "Algebraic Number Theory," especially pages 3–4 and the bottom of page 7.
- As a generalization of the previous example, let R be a commutative ring and let p be a prime ideal of R. Then R - p is a multiplicative system and the corresponding localization is denoted Rp. The unique maximal ideal is then p.
- Let R be a commutative ring and f an element of R. we can consider the multiplicative system {fn : n = 0,1,...}. Then the localization intuitively is just the ring obtained by inverting powers of f. If f is nilpotent, the localization is the zero ring.
Properties
Some properties of the localization R* = S −1R:- S−1R = {0} if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
S contains 0. - The ring homomorphism R → S −1R is injective if and only if S does not contain any zero divisorZero divisorIn abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s. - There is a bijectionBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
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 ringLocal ringIn abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
, 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 algebraCommutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and are used to construct the rings of functions on open subsets
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
of the spectrum of a ring
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
, Spec(R).
- The set S consists of all powers of a given element r. The localization corresponds to restriction to the Zariski open subset Ur ⊂ 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 ringPolynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
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 complementComplement (set theory)In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of a given prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
P in R. The primality 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 irreducibleIrreducible componentIn mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equationis the union of the two linesandThe notion of irreducibility is stronger than connectedness.- Definition :...
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 conditionOre condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring...
.
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 equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. There is now a large mathematical theory about it, named microlocalization
Microlocal analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations...
, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.