DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

total ring of fractions

Definition

For a nonzero commutative ring $R$ , let $S$ be the set of its nonzerodivisors, meaning elements whose multiplication maps on $R$ are injective. Its total ring of fractions is $Q (R) = S^{- 1} R$ . The set $S$ is multiplicative since composites of injective multiplication maps are injective. The natural map $R \to Q (R)$ is injective: $a / 1 = 0$ implies $s a = 0$ for some $s \in S$ , hence $a = 0$ . Set $Q (0) = 0$ . For a domain this recovers the fraction field; for a ring with zero divisors it need not be a field.

Depends on

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

Used by

Dependency tree · two levels

4 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

10.37.16 proof, total fraction ring used there (standard reference, not scraped)