Localisation (ring theory)

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This editable Main Article is under development and subject to a disclaimer.

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

## Construction[edit]

Let R be a commutative ring and S a non-empty subset of R closed under multiplication. The localisation  S − 1 R {\displaystyle S^{-1}R} is an R-algebra in which the elements of S become invertible, constructed as follows. Consider the set  R × S {\displaystyle R\times S} with an equivalence relation ( x , s ) ∼ ( y , t ) ⇔ x t = y s {\displaystyle (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 {\displaystyle S^{-1}R} under the operations

x s + y t = x t + y s s t {\displaystyle {\frac {x}{s}}+{\frac {y}{t}}={\frac {xt+ys}{st}}}
x s ⋅ y t = x y s t . {\displaystyle {\frac {x}{s}}\cdot {\frac {y}{t}}={\frac {xy}{st}}.}

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

### Localisation at a prime ideal[edit]

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

## Field of fractions[edit]

If R is an integral domain, then the non-zero elements  S = R ∖ { 0 } {\displaystyle 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.