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

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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

1.1

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

L1given

2.1

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 $D (1 - e) = {k \times (0)}$ .

A1step 1.1algebra

3.1

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

step 2.1∎

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

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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)