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.

The ideal generated by a subset and principal ideals

Definition

The ideal generated by a subset and principal ideals.

For $S \subseteq R$ , define $(S)$ to be the intersection of all two-sided ideals of $R$ containing $S$ . This family is nonempty because it contains $R$ , and Ideal criteria and intersections of ideals shows that $(S)$ is an ideal. For $a \in R$ , the ideal $({a})$ is written $(a)$ and is called principal.

Depends on

Dependency tree · 2 levels

Sources

Janssen and Lindsey, Rings with Inquiry, Ideals