Let $M$ be a finitely generated (not required) module over a Noetherian ring $R$. I've learned two definitions of associated primes:
$P\subset R$ is an associated prime ideal of $M$ if $P$ is the annihilator of some element of $M$.
$P\subset R$ is an associated prime of $M$ if it is the generic point of an irreducible component of the support in $\operatorname{Spec}R$ of some $m\in M$.
The second definition seems to be the same as saying $P$ is minimal among primes containing the annihilator of $m$. This makes it very close, but not quite the same as the first definition.
If P is minimal over the annihilator of some $m$, why must $P$ be precisely the annihilator of some $m'$?