Alphabeta MathHow 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.
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- 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 is left Noetherian when its left regular module is Noetherian, and right Noetherian when the right regular module is Noetherian. Unqualified “Noetherian ring” means left Noetherian here; the side is stated whenever both notions occur.
Depends on
Used by
- Every principal ideal domain is Noetherian Corollary
- Dedekind domains Definition
- embedding dimension and regular local ring Definition
- Finite graded Aₘ-modules, internal shifts and the vertex projectives Definition
- Regular points of locally Noetherian schemes Definition
- 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
- A field has only the zero ideal and itself, hence is Noetherian Lemma
- A nonzero module over a Noetherian ring has a maximal element annihilator Lemma
- All initial forms define the tangent cone Lemma
- Finite-variable polynomial algebras over fields are Noetherian by finite generators Lemma
- Local flatness criterion by regular parameters Lemma
- Local Koszul Acyclicity Inductive Converse Lemma
- Local Koszul H One Detects First Regularity Failure Lemma
- Polynomial rings over normal domains are normal Lemma
- Submodules of finite modules over a Noetherian ring are finite by induction Lemma
- The graded horseshoe lemma for finite graded projective resolutions Lemma
- The scheme-theoretic linear span of the tangent cone 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 commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member Theorem
- A Noetherian ring is Artinian exactly when every prime ideal is maximal Theorem
- A Noetherian valuation ring is a field or a DVR Theorem
- Adic completion is exact on finite modules over a Noetherian ring Theorem
- Completion of a Noetherian ring is Noetherian Theorem
- Equivalent characterizations of a DVR Theorem
- Every commutative Artinian ring is Noetherian Theorem
- Finitely generated modules over a left Noetherian ring are Noetherian Theorem
- Height-one localizations of normal Noetherian domains are DVRs Theorem
- Polynomial algebras over fields have finite integral closures Theorem
- Regular equals smooth over a perfect field Theorem
Dependency tree · two levels
6 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
- Keith Conrad, Noetherian Modules, Sections 1-2 (standard reference, not scraped)