Let $M$ be a finitely generated module over a Noetherian ring $A$. Suppose that $p\subseteq A$ is a prime ideal which is contained in the support of $M$, i.e., $M_p\neq 0$. Then does there always exist a prime chain of $M$, i.e., a finite sequence of the form $M_0(=0)\subseteq M_1\subseteq \dots \subseteq M_n(=M)$ where each successive quotient is isomorphic to $A/p_i$ for some prime ideal $p_i\subseteq A$, such that $p=p_j$ for some $j$?
As for motivation, if $M_0(=0)⊆M_1⊆\cdots⊆M_n(=M)$ is a prime chain of $M$ with primes $\{p_1,...,p_n\}$, then each $p_i$ is contained in the support of $M$. So I was wondering if every prime which is in the support of $M$ appears in some prime chain of $M$? Of course, with the support being not finite (usually), we cannot expect all of them to appear in any particular prime chain. But at least, for each one can we find a prime chain where it appears?
OK, got it. If $p$ is a prime in the support of $M$, then the annihilator of $M/pM$ is $p$. Since $p$ is associated to $M/pM$, there exists $pM\subset M_1\subseteq M$ such that $M_1/pM\cong A/p$. Now take prime chains of $M/M_1$ and $pM$. Together, we get a prime chain of $M$ where $p$ appears as one of the primes.