Definition:Localization of Ring at Element

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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 $A[f^{-1}]$.

Also see