Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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.

A ring is Jacobson iff every prime ideal is an intersection of maximal ideals containing it

Statement

For a commutative ring R, the following are equivalent.

  1. R is Jacobson.
  2. Every prime ideal p⊆R equals the intersection of the maximal ideals of R that contain p.

Facts & Assumptions

Given: A commutative ring R.

[A1]

On this page, the term "Jacobson" is used for the prime-intersection property written in statement 2.

Proof

technique · direct
1.1

Statement 1 says that R is Jacobson. By [A1], this means exactly the property written in statement 2: every prime ideal is the intersection of the maximal ideals containing it.

A1givenalgebra
2.1

Therefore statement 1 implies statement 2 and statement 2 implies statement 1, because the two sentences are literally the same condition written once as terminology and once in expanded form.

step 1.1algebra∎

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources