PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.
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.

A fraction $r / s$ is a unit in $S^{- 1} R$ exactly when $a r \in S$ for some $a \in R$

Statement

Let $R$ be a commutative ring and $S \subseteq R$ multiplicative. A fraction $r / s \in S^{- 1} R$ is a unit if and only if $a r \in S$ for some $a \in R$ .

Facts & Assumptions

Given: A fraction $r / s$ in $S^{- 1} R$ .

[F1]

Fractions satisfy the usual multiplication law, and equality is detected by an annihilating element of $S$ (The localisation relation is an equivalence relation and fraction arithmetic is well defined, Equality, vanishing, and the kernel of the localisation map).

Proof

technique · direct

1.1

If $a r \in S$ , then $a s / (a r)$ is a valid fraction and $(r / s) (a s / (a r)) = a r s / (s a r) = 1$ by [F1]. Hence $r / s$ is a unit.

2.1

Conversely, suppose $(r / s) (b / t) = 1$ . Then $r b / (s t) = 1$ , so [F1] supplies $u \in S$ with $u (r b - s t) = 0$ . Therefore $(u b) r = u s t \in S$ , because $u, s, t \in S$ . Taking $a = u b$ proves the criterion.

F1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

The CRing Project, Chapter 13, Section 13.1 (standard reference, not scraped)

A fraction r/sr/s is a unit in S−1RS−1R exactly when ar∈Sar∈S for some a∈Ra∈R