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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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Noetherian commutative rings and modules
Definition
Let be a commutative ring (Commutative ring) and let be an -module (Unital left and right modules over a ring; unqualified module means left module).
The module is Noetherian if every submodule of (Submodule of a module) is finitely generated (Generated submodule, cyclic and finitely generated modules, module basis and free module).
The ring is Noetherian if its regular module is Noetherian. Equivalently, every ideal of is finitely generated.
Under Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain), this is equivalent to every ascending chain of submodules or ideals stabilizing. The forward implication is choice-free: the union of a nonstabilizing ascending chain cannot be finitely generated. Conversely, from a submodule that is not finitely generated, Dependent Choice successively selects an element outside the span of those already selected, producing a nonstabilizing ascending chain.
Depends on
- Commutative ring
- Unital left and right modules over a ring; unqualified module means left module
- Submodule of a module
- Generated submodule, cyclic and finitely generated modules, module basis and free module
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
- A cusp family defeats the missing source hypothesis Counterexample
- Intersection multiplicity of closed subschemes at a point Definition
- Locally Noetherian and Noetherian schemes Definition
- The intrinsic Zariski tangent space Definition
- A coherent closed-point skyscraper Example
- A non-Cohen--Macaulay local quotient Example
- A quotient by a Weierstrass polynomial is a finite module over the smaller germ ring Lemma
- Eventual generation of coherent projective twists Lemma
- Finite modules over Noetherian rings are Noetherian Lemma
- Finite-type field extensions with zero Ω Lemma
- High-degree section module is finite graded Lemma
- Noetherian approximation of proper flat finitely presented sheaf data Lemma
- Noetherian domains are atomic Lemma
- Projective coherent finiteness and large twist vanishing Lemma
- Universal finite projective cohomology complex over any base Lemma
- Tangent dimension bounds local dimension Theorem
- The ring of holomorphic germs is Noetherian Theorem
- Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem
Dependency tree · two levels
16 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
- Jaap Korevaar and Jan Wiegerinck, Several Complex Variables, Definition 4.5.8 (standard reference, not scraped)
- The Stacks Project, Tag 00DV (standard reference, not scraped)