DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions

Definition

Let $R$ be a commutative ring. A subset $S \subseteq R$ is multiplicative if $1 \in S$ and $s, t \in S$ implies $s t \in S$ .

On $R \times S$ , define $(r, s) \sim (r^{'}, s^{'}) ⟺ u (r s^{'} - r^{'} s) = 0 for some u \in S .$ The localisation of $R$ at $S$ , denoted $S^{- 1} R$ , is the set of equivalence classes for this relation. The class of $(r, s)$ is written $r / s$ , and its arithmetic is $\frac{r}{s} + \frac{r^{'}}{s^{'}} = \frac{r s^{'} + r^{'} s}{s s^{'}}, \frac{r}{s} \frac{r^{'}}{s^{'}} = \frac{r r^{'}}{s s^{'}} .$ The localisation map is the ring homomorphism $λ_{S} : R ⟶ S^{- 1} R, r ⟼ r / 1.$ Every $s \in S$ maps to a unit, with $(s / 1)^{- 1} = 1 / s$ . The construction permits $0 \in S$ ; in that case the localisation is the zero ring.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Section 10.9: Localization

Multiplicative subsets and the localisation S−1RS−1R as equivalence classes of fractions

Definition

Depends on

Dependency tree · 2 levels

Sources

Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions