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−1R is an R-algebra in which the elements of S become invertible, constructed as follows. Consider the set R×S with an equivalence relation (x,s)∼(y,t)⇔xt=ys. We denote the equivalence class of (x,s) by x/s. Then the quotient set becomes a ring S−1R under the operations

xs+yt=xt+ysst
xs⋅yt=xyst.

The zero element of S−1R is the class 0/s and there is a unit element s/s. The base ring R is embedded as x↦xss.

### Localisation at a prime ideal[edit]

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

## Field of fractions[edit]

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