LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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The module-fraction relation is an equivalence relation

Statement

For a commutative ring $R$ , a multiplicative subset $S \subseteq R$ , and a left $R$ -module $M$ , the relation on $M \times S$ from Localisation of a module at a multiplicative subset is an equivalence relation.

Facts & Assumptions

Given: A commutative ring $R$ , a multiplicative subset $S \subseteq R$ , and a left $R$ -module $M$ .

[L1]

In the localisation of a module, $(m, s) \sim (n, t)$ means that $u (t m - s n) = 0$ for some $u \in S$ , and $1 \in S$ while products of elements of $S$ stay in $S$ (Localisation of a module at a multiplicative subset).

Proof

technique · direct

1.1L1algebra

Reflexivity holds because $1 (s m - s m) = 0$ , so $(m, s) \sim (m, s)$ for every $(m, s) \in M \times S$ .

1.2L1algebra

Symmetry holds because $u (t m - s n) = 0$ implies $u (s n - t m) = 0$ , so $(m, s) \sim (n, t)$ implies $(n, t) \sim (m, s)$ .

1.3L1algebra

If $(m, s) \sim (n, t)$ via $u$ and $(n, t) \sim (p, v)$ via $w$ , then $u w v (t m - s n) = 0$ and uws $(v n - t p) = 0$ ; adding these equalities gives $u w t (v m - s p) = 0$ , so $(m, s) \sim (p, v)$ .

2.1step 1.1step 1.2step 1.3∎

Steps 1.1, 1.2, and 1.3 prove that $\sim$ is an equivalence relation.

Depends on

Localisation of a module at a multiplicative subset

Used by

Addition of localised module fractions is independent of representatives Lemma

Cited to discharge well-definedness by Localisation of a module at a multiplicative subset.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Section 12 (standard reference, not scraped)