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Localization is commonly done with respect to a [[multiplicatively closed set]] {{mvar|S}} (also called a ''multiplicative set'' or a ''multiplicative system'') of elements of a ring {{mvar|R}}, that is a subset of {{mvar|R}} that is [[closure (mathematics)|closed]] under multiplication, and contains {{math|1}}.
The requirement that {{mvar|S}} must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to {{mvar|S}}. The localization by a set {{mvar|U}} that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of {{mvar|U}}. However, the same localization is obtained by using the multiplicatively closed set {{mvar|S}} of all products of elements of {{mvar|U}}. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.
For example, the localization by a single element {{mvar|s}} introduces fractions of the form <math>\tfrac a s,</math> but also products of such fractions, such as <math>\tfrac {ab} {s^2}.</math> So, the denominators will belong to the multiplicative set <math>\{1, s, s^2, s^3,\ldots\}</math> of the powers of {{mvar|s}}. Therefore, one generally talks of "the localization by the power of an element" rather than of "the localization by an element".
The localization of a ring {{mvar|R}} by a multiplicative set {{mvar|S}} is generally denoted <math>S^{-1}R,</math> but other notations are commonly used in some special cases: if <math>S= \{1, t, t^2,\ldots \}</math> consists of the powers of a single element, <math>S^{-1}R</math> is often denoted <math>R_t;</math> if <math>S=R\setminus \mathfrak p</math> is the [[complement (set theory)|complement]] of a [[prime ideal]] <math>\mathfrak p</math>, then <math>S^{-1}R</math> is denoted <math>R_\mathfrak p.</math>
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=== Integral domains ===
When the ring {{mvar|R}} is an [[integral ___domain]] and {{mvar|S}} does not contain {{math|0}}, the ring <math>S^{-1}R</math> is a subring of the [[field of fractions]] of {{mvar|R}}.
More precisely, it is the [[subring]] of the field of fractions of {{mvar|R}}, that consists of the fractions <math>\tfrac a s</math> such that <math>s\in S.</math> This is a subring since the sum <math>\tfrac as + \tfrac bt = \tfrac {at+bs}{st},</math> and the product <math>\tfrac as \, \tfrac bt = \tfrac {ab}{st}</math> of two elements of <math>S^{-1}R</math> are in <math>S^{-1}R.</math> This results from the defining property of a multiplicative set, which implies also that <math>1=\tfrac 11\in S^{-1}R.</math> In this case, {{mvar|R}} is a subring of <math>S^{-1}R.</math> It is shown below that this is no longer true in general, typically when {{mvar|S}} contains [[zero divisor]]s.
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The universal property satisfied by <math>j\colon R\to S^{-1}R</math> is the following: if <math>f\colon R\to T</math> is a ring homomorphism that maps every element of {{mvar|S}} to a [[unit (ring theory)|unit]] (invertible element) in {{mvar|T}}, there exists a unique ring homomorphism <math>g\colon S^{-1}R\to T</math> such that <math>f=g\circ j.</math>
Using [[category theory]], this can be expressed by saying that localization is a [[functor]] that is [[left adjoint]] to a [[forgetful functor]]. More precisely, let <math>\mathcal C</math> and <math>\mathcal D</math> be the categories whose objects are [[ordered pair|pairs]] of a commutative ring and a [[submonoid]] of, respectively, the multiplicative [[semigroup]] or the group of the units of the ring. The [[morphism]]s of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let <math>\mathcal F\colon \mathcal D \to \mathcal C</math> be the forgetful functor that forgets that the elements of the second element of the pair are invertible.
Then the factorization <math>f=g\circ j</math> of the universal property defines a bijection
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== Terminology explained by the context ==
The term ''localization'' originates in the general trend of modern mathematics to study [[geometry|geometrical]] and [[topology|topological]] objects ''locally'', that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of [[manifold]]s, [[germ (mathematics)|germs]] and [[sheaf (mathematics)|sheafs]]. In [[algebraic geometry]], an [[affine algebraic set]] can be identified with a [[quotient ring]] of a [[polynomial ring]] in such a way that the points of the algebraic set correspond to the [[maximal ideal]]s of the ring (this is [[Hilbert's Nullstellensatz]]). This correspondence has been generalized for making the set of the [[prime ideal]]s of a [[commutative ring]] a [[topological space]] equipped with the [[Zariski topology]]; this topological space is called the [[spectrum of a ring|spectrum of the ring]].
In this context, a ''localization'' by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as ''points'') that do not intersect the multiplicative set.
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:<math> r\, \frac{m}{s} = \frac r1 \frac ms = \frac{rm}s.</math>
It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions.
The localization of a module can be equivalently defined by using [[tensor product of modules|tensor products]]:
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is an isomorphism. If <math>M</math> is a [[finitely presented module]], the natural map
:<math>S^{-1} \operatorname{Hom}_R (M, N) \to \operatorname{Hom}_{S^{-1}R} (S^{-1}M, S^{-1}N)</math>
is also an isomorphism.{{
If a module ''M'' is a [[finitely generated module|finitely generated]] over ''R'', one has
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==Localization at primes==
The definition of a [[prime ideal]] implies immediately that the [[set complement|complement]] <math>S=R\setminus \mathfrak p</math> of a prime ideal <math>\mathfrak p</math> in a commutative ring {{mvar|R}} is a multiplicative set. In this case, the localization <math>S^{-1}R</math> is commonly denoted <math>R_\mathfrak p.</math> The ring <math>R_\mathfrak p</math> is a [[local ring]], that is called ''the local ring of {{mvar|R}}'' at <math>\mathfrak p.</math> This means that <math>\mathfrak p\,R_\mathfrak p=\mathfrak p\otimes_R R_\mathfrak p</math> is the unique [[maximal ideal]] of the ring <math>R_\mathfrak p.</math>
Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of [[Nakayama lemma]]. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is [[regular ring|regular]] if and only if all its local rings are [[regular local ring]]s.
Properties of a ring that can be characterized on its local rings are called ''local properties'', and are often the algebraic counterpart of geometric [[local property|local properties]] of [[algebraic varieties]], which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see {{slink||Localization to Zariski open sets}}, below.)
Many local properties are a consequence of the fact that the module
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*Atiyah and MacDonald. Introduction to Commutative Algebra. Addison-Wesley.
*[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. {{ISBN|0-387-97370-2}}.
* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=
* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1|mr=1322960 | year=1995 | volume=150}}
*Matsumura. Commutative Algebra. Benjamin-Cummings
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