noetherian

• Noetherian

From Emmy Noether.

IPA^(key): /noʊˈθɪəriən/

noetherian (not comparable)

1. (algebra) Satisfying some (usually ascending) chain condition:
   1. (of a ring) In which any ascending chain of ideals eventually becomes constant.
   2. (of a module) In which any ascending chain of submodules eventually becomes constant.
   3. (of a group) In which any ascending chain of subgroups eventually becomes constant.
   4. (of a topological space) In which any ascending chain of open subsets eventually becomes constant.
2. (mathematics, of a binary relation on a set) Such that the converse relation is well-founded (on that set).
3. (algebraic geometry, of a scheme) admitting a finite covering by open affine subsets ${\displaystyle \operatorname {Spec} A_{i}}$, where each ${\displaystyle A_{i}}$ is a noetherian ring;

• Artinian
• ascending chain condition