Looking for a special class of ideals such that If every ascending chain of ideals from this class stabilizes, then $R$ is a Noetherian ring.

Solution 1:

Claim: If every ascending chain of finitely generated ideals of $R$ stabilizes, then $R$ is Noetherian.

Proof: Let $I$ be an ideal of $R$. Setting $I_0 = (0)$, let us inductively construct ideals $I_n$ as follows. If $I_{n-1} \subsetneq I$, then choose $x_n \in I \setminus I_{n-1}$, and define $I_n = I_{n-1}+(x_n) = (x_1,\dots,x_n)$. We obtain a strictly ascending chain of finitely generated ideals $$0 \subsetneq (x_1) \subsetneq (x_1,x_2) \subsetneq \cdots.$$ By hypothesis this must terminate, so that some $(x_1,\dots,x_n) = I$. So $I$ is finitely generated, proving that $R$ is Noetherian.