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.
- 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.
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.
For later contradiction arguments, we also use the standard equivalent reformulation: a module or ring is Noetherian exactly when every ascending chain of submodules or ideals stabilizes.
Depends on
Used by
Dependency tree · two levels
10 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)