DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

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.

Prime ideals and maximal ideals in a commutative ring

Definition

Prime ideals and maximal ideals in a commutative ring.

Let $R$ be a commutative ring. A proper ideal $P ⊊ R$ is prime when $a b \in P$ implies $a \in P$ or $b \in P$ . A proper ideal $M ⊊ R$ is maximal when there is no proper ideal strictly between $M$ and $R$ ; equivalently, it is a maximal element of the poset of proper ideals ordered by inclusion.

Depends on

Dependency tree · 2 levels

Sources

Ernst, An Inquiry-Based Approach to Abstract Algebra, Maximal and Prime Ideals