Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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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.

2Z2\mathbb Z is an ideal of Z\mathbb Z but is not a subring under the library's unital convention

Example

2Z2\mathbb Z is an ideal of Z\mathbb Z but is not a subring under the library's unital convention.

Facts & Assumptions

Given: The subset 2Z⊆Z2\mathbb Z\subseteq\mathbb 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 1R1_R and closed under addition, additive inverses and multiplication).

[L4]

Z\mathbb Z is a ring with identity 11 (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 2Z2\mathbb Z is an ideal.

L1L2L3L4givenalgebra
2.1

The identity 11 does not belong to 2Z2\mathbb Z.

step 1.1L1L2L3L4given
3.1

Thus the missing identity rules out 2Z2\mathbb Z as a unital subring.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources