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localization of a commutative ring

Idea

The localization of a commutative ring $R$ at a set $S$ of its elements is a new ring $R[S{]}^{-1}$ in which the elements of $S$ become invertible (units) and which is universal with this property.

When interpreting a ring under Isbell duality as the ring of functions on some space $X$, then localizing it at $S$ corresponds to restricting to the subspace $Y\hookrightarrow X$ on which the elements in $S$ vanish.

See also commutative localization and localization of a ring (noncommutative).

Definition

Let $R$ be a commutative ring. Let $S\hookrightarrow U(R)$ be a subset of the underlying set.

Definition

The localization $L_S:R\to R[S^{-1}]$ is a homomorphism to another commutative ring $R[S^{-1}]$ such that

1. for all elements $s\in S\hookrightarrow R$ the image $L_S(s)\in R[S^{-1}]$ is invertible (is a unit);

2. for every other homomorphism $R\to \tilde{R}$ with this property, there is a unique homomorphism $R[S^{-1}]\to \tilde{R}$ such that we have a commuting diagram

   $$\begin{array}{ccc}R& \stackrel{L_S}{\to }& R[S^{-1}]\\ & \searrow & \downarrow \\ & & \tilde{R}\end{array}.$$\array{ R &\stackrel{L_S}{\to}& R[S^{-1}] \\ & \searrow & \downarrow \\ && \tilde R } \,.

Related concepts

• localization of a module

• localization of a ring, localization of a category