# Definition:Localization of Ring at Element

## 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 $\{1, f, f^2, \ldots\}$:

$A_f = (\{1, f, f^2, \ldots\})^{-1}A$

## Also denoted as

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