# 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 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.