Question on nonzerodivisors of a localized module and associated primes (Lemma 4.9 The Geometry of Syzygies)

Question

In Eisenbud, The Geometry of Syzygies, Lemma $4.9$ it's said "...$x$ is a nonzerodivisor on the localized module $M_p$ for all primes $p\neq \mathfrak{m}$. for this, it suffices that $x$ not be contained in any associated prime ideal of $M$ except possibly $\mathfrak{m}$". Why it suffices that $x$ not be contained in any associated prime ideal of $M$ except possibly $\mathfrak{m}$"?