Localization of a Commutative Ring

Overview

Localization constructs a ring in which selected elements admit inverses. It generalizes the passage from integers to rational numbers and makes algebra near a prime ideal accessible. The general construction must accommodate zero divisors; ordinary cross multiplication alone is insufficient.

Prerequisites

Use commutative rings with identity, ideals, and homomorphisms preserving identity. A multiplicative subset $S$ contains $1$ and is closed under multiplication. We allow $0\in S$, in which case localization produces the zero ring. For a nonzero localization one usually assumes $0\notin S$.

Definition and conventions

The localization $S^{-1}R$ consists of pairs $(a,s)\in R\times S$, written $a/s$, modulo

Addition and multiplication follow the usual fraction formulas. Its defining property is universal: any homomorphism $f:R\to A$ for which every $f(s)$ is a unit factors uniquely through $R\to S^{-1}R$, with $a/s$ mapped to $f(a)f(s)^{-1}$.

For a prime ideal $\mathfrak p$, write $R_{\mathfrak p}$ for localization at $R\setminus\mathfrak p$. This notation differs from inverting the elements inside $\mathfrak p$.

Intuition

Fractions enforce specified inverses with no extra arbitrary choices. At a prime ideal, elements outside the prime become invertible, while elements inside it retain their local significance. Some elements may disappear if an allowed denominator annihilates them.

Examples and non-examples

For $R=\mathbb Z$ and $S=\{1,2,4,\ldots\}$, the localization is $\mathbb Z[1/2]$. It contains $3/8$ but not $1/3$: an equation $a/2^k=1/3$ would require $3a=2^k$.

In $R=\mathbb Z/6\mathbb Z$, invert the class of $2$. Since $2\cdot3=0$, the image of $3$ becomes zero. The resulting ring is isomorphic to $\mathbb Z/3\mathbb Z$. Thus localization is not always an embedding. This example explains the factor $u$ in the equivalence relation.

The field of fractions of a domain arises by inverting every nonzero element. For a ring with zero divisors that set need not even be multiplicatively closed.

Important properties

The kernel of $R\to S^{-1}R$ consists of elements annihilated by some member of $S$. Prime ideals of the localization correspond to prime ideals of $R$ disjoint from $S$, via extension and contraction.

The ring $R_{\mathfrak p}$ is local, with maximal ideal $\mathfrak pR_{\mathfrak p}$. Any fraction whose numerator is outside $\mathfrak p$ has an explicit inverse; the remaining fractions form the unique maximal ideal. Localization of modules is exact, so it preserves short exact sequences even when the original ring map is not injective.

Connections and further reading

Localization gives the local rings attached to points of an affine scheme. It also detects module properties prime by prime. Read Milne, Section 5, Rings of fractions, for the construction and universal property.