CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

Outside a domain, the nonzero elements need not be multiplicative: $2 \cdot 3 = 0$ in $Z / 6$

Statement refuted

For every nonzero commutative ring $R$ , the subset $R ∖ {0}$ is multiplicative and can be used to define a field of fractions.

Facts & Assumptions

Given: The quotient ring $R = Z / 6 Z$ .

[F1]

A multiplicative subset must be closed under products (Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions).

[F2]

The ring $Z / 6$ is the quotient ring $Z / 6 Z$ (For every $n \in N$ , the congruence-class ring $Z / n$ is the quotient ring $Z / n Z$ ).

[F3]

The field-of-fractions construction at all nonzero elements is defined for integral domains (The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain).

Counterexample

technique · direct

1.1

In $R = Z / 6 Z$ , the classes of $2$ and $3$ are nonzero, but their product is the class of $6$ , hence zero.

F2algebra

2.1

Therefore $R ∖ {0}$ is not closed under multiplication and is not multiplicative by [F1]. This shows why the domain hypothesis in [F3] cannot be removed.

F1F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

The Stacks Project, Section 10.9: Localization (standard reference, not scraped)

Outside a domain, the nonzero elements need not be multiplicative: 2⋅3=02⋅3=0 in Z/6Z/6