DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Localisation of a module at a multiplicative subset

Definition

Let $R$ be a commutative ring, let $S \subseteq R$ be a multiplicative subset, and let $M$ be a left $R$ -module. On $M \times S$ , define $(m, s) \sim (n, t) ⟺ u (t m - s n) = 0 for some u \in S .$

The localisation of $M$ at $S$ is the set $S^{- 1} M$ of equivalence classes for this relation, and the class of $(m, s)$ is written $m / s$ .

The proposed addition and $S^{- 1} R$ -scalar action are $\frac{m}{s} + \frac{n}{t} : = \frac{t m + s n}{s t}, \frac{a}{u} \cdot \frac{m}{s} : = \frac{a m}{u s} .$ The canonical map is $λ_{M} : M ⟶ S^{- 1} M, m ⟼ \frac{m}{1} .$

The relation, the addition formula, and the scalar action are justified by The module-fraction relation is an equivalence relation ↗, Addition of localised module fractions is independent of representatives ↗, and The localised scalar action is independent of representatives ↗.

Localisation of a module at a multiplicative subset

Definition

Depends on

Dependency tree · 2 levels

Sources