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 $$S^{-1}R$$ is an R-algebra in which the elements of S become invertible, constructed as follows. Consider the set $$R \times S$$ with an equivalence relation $$(x,s) \sim (y,t) \Leftrightarrow xt = ys$$. We denote the equivalence class of (x,s) by x/s. Then the quotient set becomes a ring $$S^{-1}R$$ under the operations


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

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

Localisation at a prime ideal
If $$\mathfrak{p}$$ is a prime ideal of R then the complement $$S = R \setminus \mathfrak{p}$$ is a multiplicatively closed set and the localisation of R at $$\mathfrak{p}$$ is the localisation at S.

Field of fractions
If R is an integral domain, then the non-zero elements $$S = R \setminus \{0\}$$ 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.