Localisation (ring theory)

In ring theory, the localisation of a ring is an extension ring in which elements of the base ring become invertible.

Construction

Let R be a commutative ring and S a non-empty subset of R closed under multiplication. The localisation $S^{-1}R$ is an R-algebra in which the elements of S become invertible, constructed as follows. Consider the set $R\times S$ with an equivalence relation $(x,s)\sim (y,t)\Leftrightarrow xt=ys$. We denote the equivalence class of (x,s) by x/s. Then the quotient set becomes a ring $S^{-1}R$ under the operations

${\frac {x}{s}}+{\frac {y}{t}}={\frac {xt+ys}{st}}$

${\frac {x}{s}}\cdot {\frac {y}{t}}={\frac {xy}{st}}.$

The zero element of $S^{-1}R$ is the class $0/s$ and there is a unit element $s/s$. The base ring R is embedded as $x\mapsto {\frac {xs}{s}}$.

Localisation at a prime ideal

If ${\mathfrak {p}}$ is a prime ideal of R then the complement $S=R\setminus {\mathfrak {p}}$ is a multiplicatively closed set and the localisation of R at ${\mathfrak {p}}$ is the localisation at S, also denoted by $R_{\mathfrak {p}}$. It is a local ring with a unique maximal ideal — the ideal generated by ${\mathfrak {p}}$ in $R_{\mathfrak {p}}$.

Field of fractions

If R is an integral domain, then the non-zero elements $S=R\setminus \{0\}$ form a multiplicatively closed subset. The localisation of R at S is a field, the field of fractions of R. A ring can be embedded in a field if and only if it is an integral domain.