Relation between associated primes and primary decomposition for non-finite modules Theorem 6.8(ii) p.41 in Matsumura's Commutative Ring Theory, says that if $A$ is a Noetherian ring, $M$ a finite $A$-module and $N=N_1 \cap \cdots \cap N_s$ an irredundant primary decomposition of a proper submodule $N$ with $Ass(M/N_i) = \left\{P_i \right\}$, then $Ass(M/N) = \left\{P_1, \cdots, P_s \right\}$, where $Ass(\cdot)$ stands for "associated primes".
According to my understanding, the proof does not use the fact that $M$ is a finite $A$-module.
I would like to confirm that the statement of theorem 6.8(ii) is valid even if $M$ is not finite over $A$ (however $A$ must still be Noetherian).
 A: $\DeclareMathOperator{\Ass}{Ass}$
Everything follows from two facts.  The first is the "short exact sequence property" of associated primes.  That is, given a short exact sequence of $A$-modules, the $\Ass$ of the left module is contained in the $\Ass$ of the middle, and the $\Ass$ of the middle is contained in the union of the $\Ass$'es of the two sides.  The second is the fact that any nonzero $A$-module has an associated prime (which in turn requires that $A$ be Noetherian).
First, to simplify notation, we may mod out by $N$ to assume that $N=0$, so that what you really have is an irredundant primary decomposition of the zero module.
Next, there is a natural homomorphism  $$M \hookrightarrow \bigoplus_{i=1}^s M/N_i$$ that is injective since its kernel is $\bigcap_{i=1}^s N_i = 0$, and hence $\Ass M \subseteq \Ass \left(\bigoplus_{i=1}^s M/N_i\right) = \bigcup_{i=1}^s \Ass(M/N_i) = \{P_1, \ldots, P_s\}$ (where the first equality follows from induction on the short exact sequence property).
For the reverse containment, by symmetry it is enough to show that $P_1 \in \Ass M$.  Consider the following composition of natural homomorphisms: $$
N_2 \cap \cdots \cap N_s \hookrightarrow M \twoheadrightarrow M/N_1.
$$
This composition is an injection.  Hence, $\Ass (N_2 \cap \cdots \cap N_s) \subseteq \Ass(M/N_1) = \{P_1\}$.  But $N_2 \cap \cdots \cap N_s \neq 0$ (by irredundancy) and since any nonzero $A$-module has an associated prime, $\Ass (N_2 \cap \cdots \cap N_s) = \{P_1\}$.  But since $N_2 \cap \cdots \cap N_s$ is a submodule of $M$, it follows that $P_1 \in \Ass M$.
