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 modules: every submodule is finitely generated
Definition
A left -module is Noetherian when every submodule of is finitely generated (Generated submodule, cyclic and finitely generated modules, module basis and free module). This finite-generation definition is the convention; its equivalence with ACC and the maximal condition is proved in Finite generation, ACC, and maximal-condition characterizations of Noetherian modules.
Depends on
Used by
- Left and right Noetherian rings Definition
- The Prüfer p-group is Artinian but not Noetherian Example
- ℤ is Noetherian but not Artinian as a module over itself Example
- Finite generation, ACC, and maximal-condition characterizations of Noetherian modules Theorem
- Noetherian and Artinian conditions are each exact in short exact sequences Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 14 results over 9 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)