Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Radicals and quotient correspondence

Statement

Let R be a commutative ring, let I⊆J⊴R be ideals, and write π:R→R/I for the quotient map. Then

J/I=J/I

as ideals of R/I. In particular, J/I is radical in R/I if and only if J is radical in R.

Facts & Assumptions

Given: A commutative ring R, ideals I⊆J⊴R, and the quotient map π:R→R/I.

[L1]

Ideals of R/I correspond to ideals of R containing I, so J/I is an ideal of R/I (Correspondence theorem: ideals of R/I correspond to ideals of R containing I).

[L2]

An element belongs to the radical of an ideal exactly when one of its positive powers lies in that ideal (The radical of an ideal).

Proof

technique · direct
1.1

Let x+I∈R/I. By [L2], x+I∈J/I exactly when (x+I)n=xn+I∈J/I for some n≥1, and that happens exactly when xn∈J. Applying [L2] again shows that this is equivalent to x∈J, so x+I∈J/I exactly when x+I∈J/I.

L1L2givenalgebra
2.1

Step 1.1 proves J/I=J/I. Consequently, J/I is radical exactly when J/I=J/I, exactly when J/I=J/I, and exactly when J=J.

step 1.1L1algebra
3.1

The quotient radical is therefore exactly the quotient of the radical, and radical ideals correspond across the quotient map.

step 1.1step 2.1∎

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