Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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.

Radical membership via positive powers

Statement

Let R be a commutative ring, let I⊴R be an ideal, and let x∈R. Then

x∈I⟺xn∈I for some integer n≥1.

In particular, R=R, and if R is the zero ring then (0)=R.

Facts & Assumptions

Given: A commutative ring R, an ideal I⊴R, and an element x∈R.

[L1]

The radical of I is the set of elements whose positive powers land in I (The radical of an ideal).

Proof

technique · direct
1.1

The displayed equivalence is exactly the defining membership criterion in [L1].

L1
2.1

Taking I=R in step 1.1 gives R=R, because every element already has its first power in R. If R is the zero ring, then (0)=R, so the same observation gives (0)=R.

step 1.1givenalgebra
3.1

Steps 1.1 and 2.1 prove the claim and record the unit-ideal and zero-ring boundaries explicitly.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

3 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