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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 $S^{-1}R$ 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, also denoted by $R_{\mathfrak{p}}$. It is a local ring with a unique maximal ideal — the ideal generated by $\mathfrak{p}$ in $R_{\mathfrak{p}}$.

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.