CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-02

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.

The zero ideal of $Z$ is prime but not maximal

Statement refuted

Every prime ideal of a commutative ring is maximal.

Facts & Assumptions

Given: The zero ideal $(0)$ in $Z$ .

[L1]

A prime ideal is proper and satisfies the zero-product implication; a maximal ideal has no proper intermediate ideal (Prime ideals and maximal ideals in a commutative ring).

[L2]

The ideal criterion verifies subtraction closure and absorption (Ideal criteria and intersections of ideals).

[L3]

$Z$ is a nonzero commutative ring (The integers form a commutative ring).

[L4]

Integer cancellation implies $a b = 0$ only if $a = 0$ or $b = 0$ (The integers have no zero divisors; multiplicative cancellation).

Counterexample

technique · direct

1.1

The set $(0)$ is a proper ideal, and [L4] shows that $a b \in (0)$ implies $a \in (0)$ or $b \in (0)$ .

L1L2L3L4givenalgebra

2.1

The ideal $2 Z$ satisfies $(0) ⊊ 2 Z ⊊ Z$ .

step 1.1L1L2L3L4givenalgebra

3.1

Hence $(0)$ is prime but not maximal, refuting the statement.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

Ernst, An Inquiry-Based Approach to Abstract Algebra, Maximal and Prime Ideals (standard reference, not scraped)

The zero ideal of ZZ is prime but not maximal