Why does the support of a module have finitely many minimal elements?

Question

Let $A$ be a Noetherian ring and let $M$ be a finitely generated $M$-module. The support of $M$ is defined as: $$\operatorname{Supp}(M)=\{\mathfrak p\in\operatorname{Spec }A\colon M_{\mathfrak p}\neq 0\}$$

Why do we have that the set of minimal primes in $\operatorname{Supp}(M)$ is finite?

Answer 1

Because it is the set of minimal prime ideals which contain $\;\operatorname{Ann}_A M$, corresponding to the minimal primes of the noetherian ring $\;A/\operatorname{Ann}_A M$.