Commutative rings over which every module $M$ satisfies $\mathrm{Ass}(R/\mathrm{Ann}\; M) \subseteq \mathrm{Ass}(M)$

Question

Let $R$ be a commutative ring with unity. If $M$ is a Noetherian $R$-module, then we know that $\mathrm{Ass}(R/\mathrm{Ann}\; M) \subseteq \mathrm{Ass}(M)$.

My question is:

Can we characterize those commutative rings (with unity) $R$ such that $\mathrm{Ass}(R/\mathrm{Ann}\; M) \subseteq \mathrm{Ass}(M)$ for every $R$-module $M$ ? If this characterization is difficult, then can we at least say anything by assuming $R$ is Noetherian ?