Definition:Localization of Ring

Definition

Let $A$ be a commutative ring with unity.

Let $S \subseteq A$ be a multiplicatively closed subset of $A$.

A localization of $A$ at $S$ is a pair $\struct {A_S, \iota}$ where:

$A_S$ is a commutative ring with unity, the actual localization

$\iota: A \to A_S$ is a ring homomorphism, the localization homomorphism

such that:

$(1): \quad \map \iota S \subseteq A_S^\times$, where $A_S^\times$ is the group of units of $A_S$

$(2): \quad$ For every pair $\tuple {B, g}$ where:

$B$ is any ring with unity

$g: A \to B$ is a ring homomorphism such that $\map g S \subseteq B^\times$

there exists a unique ring homomorphism $h: A_S \to B$ such that:

$g = h \circ \iota$

This page has been identified as a candidate for refactoring of basic complexity. In particular: Pages which link to this (e.g. Localization of Ring Exists) refer to item $(2)$ as the "universal property for localization". Hence as that is an entity in its own right, it makes sense either to extract the above into their own pages, or invoke the {{Axiom}} template to implement names and some structure onto the above items. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

That is, the following diagram commutes:

$\begin{xy}\xymatrix@L+2mu@+1em {

A \ar[drdr]_*{g}
  \ar[rr]^*{\iota}

& & A_S \ar[dd]^*{\exists ! h} \\ \\ & & B }\end{xy}$

Notation

The localization of $A$ at $S$ can be written $S^{-1} A$, or $A \sqbrk {S^{-1} }$.

The notation $A_{\mathfrak p}$ is seen for the localization at a prime ideal $\mathfrak p$.

The notation $A_f$ is seen for the localization at an element $f \in A$.

Also known as

A localization of a ring is also known as a ring of fractions.

Also see

• Localization of Ring Exists
• Localization of Ring is Unique
• Definition:Localization of Module
• Definition:Localization of Algebra

Special cases

• Definition:Localization of Ring at Element
• Definition:Localization of Ring at Prime Ideal
• Definition:Field of Fractions
• Definition:Total Ring of Fractions

Linguistic Note

The word localization is spelt localisation in non-US English.

Sources

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