Associate prime ideals and exact sequences of $R$-modules Let $R$ be a commutative unitary ring, consider the exact sequence of $R$ - modules
$$
0\rightarrow N\rightarrow M\rightarrow L\rightarrow0.
$$
We know that 
$$\text{Ass}(M)\subseteq \text{Ass}(N)\cup \text{Ass}(L).$$
My question is: 

When (in what conditions on the ring $R$) do we have the equality
  $$
\text{Ass}(M)=\text{Ass}(N)\cup \text{Ass}(L)?
$$
  Can someone help me.

 A: Recall that a prime ideal $\mathfrak p$ of $R$ is an associated prime of a module $M$ if and only if $M$ contains a submodule isomorphic to $R/\mathfrak p$. Equivalently, there exists $x\in M$ such that $\mathfrak p=\mathrm{Ann}_R(x)$. 
So we always have $\mathrm{Ass}(N)\subseteq \mathrm{Ass}(M)$. So the only problem is to see when $\mathrm{Ass}(L)\subseteq \mathrm{Ass}(M)$ holds. 
If the Krull dimension $\dim R>0$ (equivalent to non-artinian if $R$ is noetherian), then there are distinct prime ideals $\mathfrak p\subset \mathfrak m$. Consider the canonical surjection 
$$M:=R/\mathfrak p\to L:=R/\mathfrak m.$$ 
Then $\{\mathfrak m \}=\mathrm{Ass}(L) \not\subset \{\mathfrak p\}=\mathrm{Ass}(M)
$. 
So the rings $R$ for which the desired equality holds for all exact sequences are dimension zero. 
Edit 2. The above conclusion is frustrating because it doesn't give any positive information when $\dim R>0$. However:

Suppose $R$ is noetherian. Consider $\mathrm{Spec}(R)$ with its Zariski topology. Then 
  $$\mathrm{Ass}(M)\subseteq \mathrm{Ass}(N)\cup \mathrm{Ass}(L)\subseteq \overline{\mathrm{Ass}(M)}.$$

Proof. Let $\mathfrak p\in \mathrm{Ass}(L)$. We have to show that $\mathfrak p$ contains a $\mathfrak q\in \mathrm{Ass}(M)$. We can replace $L$ by a suitable submodule (and $M$ by the preimage of this submodule) and suppose $L$ is isomorphic to $R/\mathfrak p$. Suppose that  $\mathfrak p$ doesn't contain any associated prime of $M$. Then the localization $M_{\mathfrak p}$ (viewed as an $R$-module) has no associated prime (Matsumura: Commutative algebra, Lemma (7.C)). So $M_{\mathfrak p}=0$ (op. cit., Cor. 1, p. 50). Hence for any $x\in M$, there exists $a\in R\setminus \mathfrak p$ with $ax=0$. So $a\phi(x)=0$ and $\phi(x)=0$, $M=\mathrm{ker}(\phi)$. Impossible. 
