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.