ExampleConstruction: AI-adaptedVerification: 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.

$2 Z$ is an ideal of $Z$ but is not a subring under the library's unital convention

Example

$2 Z$ is an ideal of $Z$ but is not a subring under the library's unital convention.

Facts & Assumptions

Given: The subset $2 Z \subseteq Z$ .

[L1]

A two-sided ideal is an additive subgroup with multiplication absorption (Left, right and two-sided ideals).

[L2]

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

[L3]

A subring contains the ambient identity (Subring: a subset containing $1_{R}$ and closed under addition, additive inverses and multiplication).

[L4]

$Z$ is a ring with identity $1$ (The integers form a commutative ring).

Verification

technique · direct

1.1

Differences of even integers are even, and multiplying an even integer by any integer remains even, so $2 Z$ is an ideal.

L1L2L3L4givenalgebra

2.1

The identity $1$ does not belong to $2 Z$ .

step 1.1L1L2L3L4given

3.1

Thus the missing identity rules out $2 Z$ as a unital subring.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)

2Z2Z is an ideal of ZZ but is not a subring under the library's unital convention