If $R=\mathbb {Z}$ is the ring of , and $S=\mathbb {Z} \setminus \{0\},$ then $S^{-1}R$ is the field $\mathbb {Q}$ of the rational numbers.

integers

If $R$ is an , and $S=R\setminus \{0\},$ then $S^{-1}R$ is the field of fractions of $R$ . The preceding example is a special case of this one.

integral domain

If $R$ is a , and if $S$ is the subset of its elements that are not zero divisors, then $S^{-1}R$ is the total ring of fractions of $R$ . In this case, $S$ is the largest multiplicative set such that the homomorphism $R\to S^{-1}R$ is injective. The preceding example is a special case of this one.

commutative ring

If $x$ is an element of a commutative ring $R$ and $S=\{1,x,x^{2},\ldots \},$ then $S^{-1}R$ can be identified (is to) $R[x^{-1}]=R[s]/(xs-1).$ (The proof consists of showing that this ring satisfies the above universal property.) This sort of localization plays a fundamental role in the definition of an affine scheme.

canonically isomorphic

If ${\mathfrak {p}}$ is a of a commutative ring $R$ , the set complement $S=R\setminus {\mathfrak {p}}$ of ${\mathfrak {p}}$ in $R$ is a multiplicative set (by the definition of a prime ideal). The ring $S^{-1}R$ is a local ring that is generally denoted $R_{\mathfrak {p}},$ and called the local ring of $R$ at ${\mathfrak {p}}.$ This sort of localization is fundamental in commutative algebra, because many properties of a commutative ring can be read on its local rings. Such a property is often called a local property. For example, a ring is regular if and only if all its local rings are regular.

prime ideal

The multiplicative set is the of a prime ideal ${\mathfrak {p}}$ of a ring $R$ . In this case, one speaks of the "localization at ${\mathfrak {p}}$ ", or "localization at a point". The resulting ring, denoted $R_{\mathfrak {p}}$ is a local ring, and is the algebraic analog of a ring of germs.

complement

The multiplicative set consists of all powers of an element $t$ of a ring $R$ . The resulting ring is commonly denoted $R_{t},$ and its spectrum is the Zariski open set of the prime ideals that do not contain $t$ . Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a consisting of Zariski open sets of this form).

neighborhood basis

The term localization originates in the general trend of modern mathematics to study geometrical and topological objects locally, that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of manifolds, germs and 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 ideals of the ring (this is Hilbert's Nullstellensatz). This correspondence has been generalized for making the set of the prime ideals of a commutative ring a topological space equipped with the Zariski topology; this topological space is called the 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.

Two classes of localizations are more commonly considered:

In number theory and algebraic topology, when working over the ring $\mathbb {Z}$ of integers, one refers to a property relative to an integer $n$ as a property true at $n$ or away from $n$ , depending on the localization that is considered. "Away from $n$ " means that the property is considered after localization by the powers of $n$ , and, if $p$ is a prime number, "at $p$ " means that the property is considered after localization at the prime ideal $p\mathbb {Z}$ . This terminology can be explained by the fact that, if $p$ is prime, the nonzero prime ideals of the localization of $\mathbb {Z}$ are either the singleton set ${p}$ or its complement in the set of prime numbers.

$1\in S^{-1}I\quad \iff \quad 1\in \operatorname {sat} (I)\quad \iff \quad S\cap I\neq \emptyset$

$I\subseteq J\quad \ \implies \quad \ S^{-1}I\subseteq S^{-1}J\quad \ {\text{and}}\quad \ \operatorname {sat} (I)\subseteq \operatorname {sat} (J)$
(this is not always true for )

strict inclusions

$S^{-1}(I\cap J)=S^{-1}I\cap S^{-1}J,\qquad \,\operatorname {sat} (I\cap J)=\operatorname {sat} (I)\cap \operatorname {sat} (J)$

$S^{-1}(I+J)=S^{-1}I+S^{-1}J,\qquad \operatorname {sat} (I+J)=\operatorname {sat} (I)+\operatorname {sat} (J)$

$S^{-1}(I\cdot J)=S^{-1}I\cdot S^{-1}J,\qquad \quad \operatorname {sat} (I\cdot J)=\operatorname {sat} (I)\cdot \operatorname {sat} (J)$

If ${\mathfrak {p}}$ is a such that ${\mathfrak {p}}\cap S=\emptyset ,$ then $S^{-1}{\mathfrak {p}}$ is a prime ideal and ${\mathfrak {p}}=\operatorname {sat} ({\mathfrak {p}})$ ; if the intersection is nonempty, then $S^{-1}{\mathfrak {p}}=S^{-1}R$ and $\operatorname {sat} ({\mathfrak {p}})=R.$

prime ideal

Let $S$ be a multiplicative set in a commutative ring $R$ , and $j\colon R\to S^{-1}R$ be the canonical ring homomorphism. Given an ideal $I$ in $R$ , let $S^{-1}I$ the set of the fractions in $S^{-1}R$ whose numerator is in $I$ . This is an ideal of $S^{-1}R,$ which is generated by $j (I)$ , and called the localization of $I$ by $S$ .

The saturation of $I$ by $S$ is $j^{-1}(S^{-1}I);$ it is an ideal of $R$ , which can also defined as the set of the elements $r\in R$ such that there exists $s\in S$ with $sr\in I.$

Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation. In what follows, $S$ is a multiplicative set in a ring $R$ , and $I$ and $J$ are ideals of $R$ ; the saturation of an ideal $I$ by a multiplicative set $S$ is denoted $\operatorname {sat} _{S}(I),$ or, when the multiplicative set $S$ is clear from the context, $\operatorname {sat} (I).$

$P$ holds for $M$ .

$P$ holds for all $M_{\mathfrak {p}},$ where ${\mathfrak {p}}$ is a prime ideal of $R$ .

$P$ holds for all $M_{\mathfrak {m}},$ where ${\mathfrak {m}}$ is a maximal ideal of $R$ .

Non-commutative case[edit]

Localizing non-commutative rings is more difficult. While the localization exists for every set S of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved 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.

integers

integral domain

commutative ring

canonically isomorphic

prime ideal

complement

neighborhood basis

strict inclusions

prime ideal

Non-commutative case[edit]

Local analysis

Localization of a category

Localization of a topological space

Localization