DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26

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.

The Jacobson radical of a ring

Definition

For a commutative ring $R$ , the Jacobson radical is $J (R) : = ⋂_{m maximal} m .$ When $R = 0$ , there are no maximal ideals, and on this page the empty intersection is taken to be $0$ , so $J (0) = 0$ .

Depends on

Prime ideals and maximal ideals in a commutative ring

The Jacobson radical of a ring

Definition

Depends on

Dependency tree · 2 levels

Sources