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

<math>\frac{x}{s} + \frac{y}{t} = \frac{xt+ys}{st} </math>
<math>\frac{x}{s} \cdot \frac{y}{t} = \frac{xy}{st} .</math>

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

Localisation at a prime ideal

If <math>\mathfrak{p}</math> is a prime ideal of R then the complement <math>S = R \setminus \mathfrak{p}</math> is a multiplicatively closed set and the localisation of R at <math>\mathfrak{p}</math> is the localisation at S, also denoted by <math>R_{\mathfrak{p}}</math>. It is a local ring with a unique maximal ideal — the ideal generated by <math>\mathfrak{p}</math> in <math>R_{\mathfrak{p}}</math>.

Field of fractions

If R is an integral domain, then the non-zero elements <math>S = R \setminus \{0\}</math> 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.