ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Coordinate idempotents split the spectrum of a product ring into two clopen pieces

Example

Let $k$ be a field and let $R = k \times k$ . The coordinate idempotent $e = (1, 0)$ satisfies $D (e) = {(0) \times k}, D (1 - e) = {k \times (0)},$ so $Spec (R)$ splits into two clopen singletons.

Facts & Assumptions

Given: A field $k$ , the product ring $R = k \times k$ , and the idempotent $e = (1, 0)$ .

[L1]

An idempotent partitions the spectrum into complementary clopen subsets $D (e)$ and $D (1 - e)$ (An idempotent partitions the spectrum into complementary clopen subsets).

[A1]

The prime ideals of $k \times k$ are exactly $(0) \times k$ and $k \times (0)$ .

Verification

technique · direct

1.1L1given

The element $e = (1, 0)$ satisfies $e^{2} = e$ , so [L1] gives a clopen partition $Spec (R) = D (e) ⊔ D (1 - e)$ .

2.1A1step 1.1algebra

By [A1], the only prime ideals are $(0) \times k$ and $k \times (0)$ . The ideal $k \times (0)$ contains $e$ , while $(0) \times k$ does not; hence $D (e) = {(0) \times k}$ . Similarly $D (1 - e) = {k \times (0)}$ .

3.1step 2.1∎

Therefore the coordinate idempotents split the product spectrum into two clopen singleton pieces.

Depends on

An idempotent partitions the spectrum into complementary clopen subsets

Used by

Nothing in the library uses this result yet.

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

J. S. Milne, A Primer of Commutative Algebra, v4.03, Lemma 14.2 (standard reference, not scraped)
The Stacks Project, Section 10.22: Connected components of spectra (standard reference, not scraped)