# Definition:Localization of Ring

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## 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.
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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