Corollary 1

Let $M$ be a module over a Noetherian ring $A$. Then the condition $M \neq \{0\}$ is equivalent to $\operatorname{Ass}(M) \neq \varnothing$.

If $M = \{0\}$, clearly $\operatorname{Ass}(M)$ is empty (without any hypothesis on $A$). If $M \neq \{0\}$, the set of ideals of the form $\operatorname{Ann}(x)$, where $x \in M$ and $x \neq 0$, is non-empty and consists of ideals $\# A$; as $A$ is Noetherian, this set has a maximal element; then it suffices to apply Proposition 2.