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