# Definition:Localization of Ring at Element

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## Definition

Let $A$ be a commutative ring with unity.

Let $f \in A$ be an element.

The **localization of $A$ at $f$** is the localization of $A$ at the set of powers $\set {1, f, f^2, \ldots}$:

- $A_f = \paren {\set {1, f, f^2, \ldots} }^{-1}A$

## Also denoted as

To avoid confusion with completions, the **localization** of $A$ at $f$ is also denoted $A \sqbrk {f^{-1} }$.

## Also see

- Set of Powers of Ring Element is Multiplicatively Closed
- Definition:Localization of Module at Ring Element

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