Commutative rings with unity over which every non-zero module has an associated prime

Question

Let $R$ be a commutative ring with unity such that every non-zero module over $R$ has an associated prime. Then is it true that $R$ is Noetherian ? If not, then can we say something about any possible structure of $R$ at least ?

COMMENTS: Since for every proper ideal $I$ of $R$, and non-zero $x+I \in R/I$, $\mathrm{ann}_R(x+I)=(I:x)$, so if every non-zero module over $R$ has an associated prime, then for every proper ideal $I$ of $R$, $\exists x \notin I$ such that $(I:x)$ is a prime ideal, and conversely, if this holds, then for every proper ideal $I$ of $R$, $\mathrm{Ass}(R/I)$ is non-empty, and then since for every non-zero $R$-module $M$, we have $\mathrm{Ass}(R/\mathrm{ann}(m)) =\mathrm{Ass}(Rm)\subseteq \mathrm{Ass}(M)$, for every non-zero $m$ in $M$, so every non-zero $R$-module would have an associated prime. So we have proved :

For a commutative ring (with unity) $R$, TFAE :

1) for every proper ideal $I$ of $R$, $\exists x \notin I$ such that $(I:x)$ is a prime ideal.

2) $R/I$ has an associated prime for every proper ideal $I$ of $R$.

3) Every non-zero $R$-module has an associated prime.

But condition (1) still seems very complicated. Noetherian rings of course satisfy condition (1), but I don't know what other type of rings satisfy that condition.

Also asked https://math.stackexchange.com/questions/2596679/commutative-rings-with-unity-over-which-every-non-zero-module-has-an-associated

Answer 1

There exists a non-noetherian ring $R$ such that every non-zero $R$-module has an associated prime.

This in proven in Example 2.3 in P. J. Cahen, Ascending chain conditions and associated primes, Commutative ring theory (Fès, 1992), 41-46, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.

The point is that over an arbitrary commutative ring, any non-zero module has a weakly associated prime. Thus, the question is related to asking for rings over which assassins and weak assassins coincide. Clearly, noetherian rings have this property, but the aforementioned example shows that there are also other such rings. I do not know any more classes of such rings, nor about any other (published) work on these questions. See also this question.