DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-17

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.

Left and right Noetherian rings

Definition

A unital ring $R$ is left Noetherian when its left regular module $_{R} R$ is Noetherian, and right Noetherian when the right regular module $R_{R}$ is Noetherian. Unqualified “Noetherian ring” means left Noetherian here; the side is stated whenever both notions occur.

Depends on

Used by

The ring (ℤ/2)^ℕ is not Noetherian Example
False statement: every right Noetherian ring is left Noetherian False statement
False statement: every subring of a Noetherian ring is Noetherian False statement
Finitely generated modules over a left Noetherian ring are Noetherian Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, Noetherian Modules, Sections 1-2 (standard reference, not scraped)