DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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

Used by

Dependency tree · two levels

4 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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise 10.11 (standard reference, not scraped)
The Stacks Project, Section 10.19: Nakayama's Lemma (standard reference, not scraped)