What is the meaning of Noetherian?

Satisfying some (usually ascending) chain condition:

2. In which any ascending chain of ideals eventually becomes constant.
3. In which any ascending chain of submodules eventually becomes constant.
4. In which any ascending chain of subgroups eventually becomes constant.
5. In which any ascending chain of open subsets eventually becomes constant.

In which any ascending chain of ideals eventually becomes constant.

In which any ascending chain of submodules eventually becomes constant.

In which any ascending chain of subgroups eventually becomes constant.

In which any ascending chain of open subsets eventually becomes constant.

Such that the converse relation is well-founded (on that set).

admitting a finite covering by open affine subsets ${\displaystyle \operatorname {Spec} A_{i}}$, where each ${\displaystyle A_{i}}$ is a noetherian ring;

Source: wiktionary.org