If every ascending chain of primary ideals in $R$ stabilizes, is $R$ a Noetherian ring? A commutative ring $R$ is called Noetherian if 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 $I_n=I_{n+1}=I_{n+2}=\cdots$.
My question is the following: 

Does there exist a non-Noetherian ring $R$ such that every ascending chain of primary ideals stabilizes?

Remark. Note that there exists non-Noetherian ring $R$ such that every ascending chain of prime ideals stabilizes. This happens exactly when $R$ is non-Noetherian and $\operatorname{Spec}(R)$ is Noetherian topological space. See here and Exercise 12 of Chapter 6 in Introduction to Commutative Algebra by Atiyah & Macdonald. 

In Commutative Algebra a suitable construction for counterexamples is given by the idealization of an $A$-module $M$, sometimes denoted by $I_A(M)$. If one considers $R=I_{\mathbb Z}(\mathbb Q)$, then $R$ is not noetherian (since $\mathbb Q$ is not finitely generated as $\mathbb Z$-module) and it satisfies ACC on primary ideals (since these are of the following form: $p^n\mathbb Z\times\mathbb Q$ with $p$ prime and $n\ge 1$, $\{0\}\times\mathbb Q$ or $\{0\}\times\{0\}$). 
