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# Localisation (ring theory)

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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

Let R be a commutative ring and S a non-empty subset of R closed under multiplication. The localisation  is an R-algebra in which the elements of S become invertible, constructed as follows. Consider the set  with an equivalence relation . We denote the equivalence class of (x,s) by x/s. Then the quotient set becomes a ring  under the operations




The zero element of  is the class  and there is a unit element . The base ring R is embedded as .

### Localisation at a prime ideal

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

## Field of fractions

If R is an integral domain, then the non-zero elements  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.