Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)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 Artinian rings
Definition
A unital ring is left Artinian when is Artinian, and right Artinian when is Artinian. Unqualified “Artinian ring” means left Artinian here.
Depends on
Used by
- Global functions on geometrically connected and geometrically reduced proper schemes Corollary
- A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings Lemma
- An Artinian integral domain is a field Lemma
- Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
- Conventions for this development and where dependent choice and Zorn's lemma are used Remark
- A commutative ring is Artinian exactly when it has finite length as a module over itself Theorem
- A Noetherian ring is Artinian exactly when every prime ideal is maximal Theorem
- An Artinian ring has only finitely many maximal ideals Theorem
- Every prime ideal of an Artinian ring is maximal Theorem
- The nilradical of an Artinian ring is a nilpotent ideal Theorem
Dependency tree · two levels
5 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
- Arvind Nair, Algebra I, Lecture 5 (standard reference, not scraped)