- 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*" andring 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 auniversal property . The localization of "R" by "S" can be denoted by "S"^{ -1}"R" or "R"_{"S"}.**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 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 atlocal ring .In

number theory andalgebraic 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 $mathbf\{Z\}\; [\; extstyle\{frac\{1\}\{n]$-algebra). For instance, for a field, "away from "p" means "characteristic not equal to "p". $mathbf\{Z\}\; [\; extstyle\{frac\{1\}\{2]$ is "away from 2", but $mathbf\{F\}\_2$ or $mathbf\{Z\}$ 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 a multiplicative

monoid , i.e. 1 is in "S" and for "s" and "t" in "S" we also have "st" 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 thetrivial ring {0}, the localization "R*" is {0} if 0 is in "S". Otherwise, thefield 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. 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" withdenominator s coming from "S"; in contrast with the integral domain case, one can safely 'cancel' fromnumerator anddenominator only elements of "S".This construction proceeds as follows: on "R" × "S" define an

equivalence relation ~ by setting ("r"_{1},"s"_{1}) ~ ("r"_{2},"s"_{2})iff there exists "t" in "S" such that:"t"("r"

_{1}"s"_{2}− "r"_{2}"s"_{1}) = 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 aring 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**/"n**"Z**where "n" is composite is not an integral domain. When "n" is a prime power it is a finitelocal ring , and its elements are either units ornilpotent . 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**/"n**"Z**is by theChinese remainder theorem isomorphic to**Z**/"a**"Z**×**Z**/"b**"Z**. If we take "S" to consist only of (1,0) and 1 = (1,1), then the corresponding localization is**Z**/"a**"Z**.* Let $,R\; =\; mathbb\{Z\}$, the integers, and p a prime number. If $,S\; =\; mathbb\{Z\}-\; pmathbb\{Z\}$, 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.

**Properties**Some properties of the localization "R*" = "S"

^{ -1}"R":* "S"

^{-1}"R" = {0}if and only if "S" contains 0.

* The ring homomorphism "R" → "S"^{ -1}"R" is injective if and only if "S" does not contain anyzero divisor s.

* There is abijection between the set of prime ideals of "S"^{-1}"R" and the set of prime ideals of "R" which do not intersect "S". This bijection is induced by the given homomorphism "R" → "S"^{ -1}"R".

* In particular: after localization at a prime ideal "P", one obtains alocal 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 andalgebraic geometry and are used to construct the rings of functions on open subsets inZariski topology of thespectrum 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 thepolynomial ring and "r" = "X" then the localization produces the ring ofLaurent polynomial s "K" ["X", "X"^{−1}] . In this case, localization corresponds to the embedding "U" ⊂ "A"^{1}, where "A"^{1}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 fordifferential 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 withFourier theory , in particular.**References*** Serge Lang, "Analytic Number Theory," Springer, 2000. pages 3-4.

**See also***

Localization of a module

*Localization of a category

*Local ring

*Valuation ring

*Homomorphism **External links*** [

*http://mathworld.wolfram.com/Localization.html Localization*] fromMathWorld .

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