A commutative ring $R$ is called Noetherian if any one of the following holds:
$1.$ Every ascending chain of ideals in $R$ stabilizes, that is, $$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$ implies the existence of $n\in\mathbb{N}$ such that $I_n=I_{n+1}=I_{n+2}=\cdots$.
$2.$ Every ideal in $R$ is finitely generated.
But I. S. Cohen proved that for a ring to be Noetherian it suffices that every prime ideal is finitely generated. So for a ring to be Noetherian we need to check above condition $2$ only for prime ideals.
So the natural question is there any class of ideals for choosing ideals $I_i$ of Ascending Chain Condition for proving that $R$ is Noetherian?
Here are two cases which does not work:
$1.$ Every ascending chain of prime ideals terminates even $R$ is not noetherian. So we can not choose $I_i$ only from Spec($R$).
Example: $R=\dfrac{k[x_1,x_2,...]}{I^2}$, where $I=(x_1,x_2,...)$
$2$. Every chain of primary ideals terminates even $R$ is not Noetherian.
Example: $\prod_{i\in\Bbb N} F_i$ where the $F_i$ are fields.
This is clearly not Noetherian, and because it is commutative and von Neumann regular, all of its primary ideals are maximal.
Any references/ideas?