Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-31
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.

Computing an inverse fractional ideal explicitly

Example

In R=Z, let I=(2/3)Z. Then

(R:I)=(3/2)Z,

and therefore I(R:I)=R.

Facts & Assumptions

Given: The fractional ideal I=(2/3)Z of R=Z.

[F1]

The inverse candidate is defined by (R:I)={x∈Q:xI⊆R} (Products, colons, and inverse candidates for fractional ideals).

[L1]

Fractional ideals factor uniquely into prime powers (Unique factorization of nonzero fractional ideals into prime powers).

Verification

technique · direct
1.1

An element x∈Q lies in (R:I) exactly when x(2/3)∈Z, equivalently when x∈(3/2)Z. Thus (R:I)=(3/2)Z.

F1givenalgebra
2.1

Multiplying the displayed generators gives (2/3)(3/2)=1, so I(R:I)=Z. This agrees with the valuation description from [L1].

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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