Definition:Localization of Ring

Definition
Let $A$ be a commutative ring with unity.

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

Then a commutative ring with unity $A_S$ together with a ring homomorphism $\iota : A \to A_S$ is the localisation of $A$ at $S$ if:


 * 1. $\iota(S) \subseteq A_S^\times$ ($A_S^\times$ is the Group of Units of $A_S$)


 * 2. If $B$ is any ring and $g : A \to B$ is a ring homomorphism such that $g(S) \subseteq B^\times$, then there exists a unique ring homomorphism $h : A_S \to B$ such that $g = h \circ \iota$

Notation
Sometimes the localisation of $A$ at $S$ is written $S^{-1}A$, or $A[S^{-1}]$.

If $\mathfrak p$ is a prime ideal of $A$, then by the definition of a prime ideal, $S = A \backslash \mathfrak p$ is multiplicatively closed. In this case is conventional to write $A_{\mathfrak p}$ for the localisation of $A$ at $S$.

If $f \in A$ is some element, then $S = \{f^n : n \geq 0\}$ is trivially multiplicatively closed, and it is common to write $A_f$ for the localisation of $A$ at $S$.