Let $\textbf{CRing}$ be the category of commutative rings with $1$. Then, an immensely important subclass is that of the Noetherian rings, those rings which satisfy either of the equivalent conditions:
- every ascending chain of ideals terminates
- every ideal is finitely generated
What are some properties for rings which are similar to, but strictly weaker than, the Noetherian property, and also appear frequently in (research) mathematics?
For example, a property which satisfies the first two conditions might be that every ideal is countably generated. However, I doubt that it satisfies the third condition.