Show that $\text{Ass}(M)\subset \{p_1,\dots,p_n\}$ and that the minimal elements of the two sets coincide (hence only depend on $M$).

Question

Let $R$ be a Noetherian ring, $M\neq 0$ a finite $R$-module, and $0=M_0\subset M_1\subset \dots\subset M_n=M$ a chain of submodules with $M_i/M_{i-1}\cong R/p_i$, $p_i\in\text{Spec}(R)$.

1. Show that $\text{Ass}(M)\subset \{p_1,\dots,p_n\}$ and that the minimal elements of the two sets coincide (hence only depend on $M$).

2. Let $p$ be minimal in $\{p_1,\dots,p_n\}$. Show that in any chain as above, the multiplicity with which the factor $R/p$ appears is $\ell_{R_P}(M_P)$ (hence only depends on $M$).

In Dummit and Foote, there is a question in Section 15.5 number 26, which seems familiar.

Suppose $M$ is a finitely generated module over the Noetherian ring $R$.

1. Prove that there are finitely many minimial primes $P_1,\dots,P_n$ containing $Ann(M)$

2. Prove that $\{P_1,\dots,P_n\}$ is also the set of minimal primes in $Ass_R(M)$ and that $Supp(M)$ is the union of the Zariski closed sets $Z(P_1),\dots,Z(P_n)$ in Spec(R).

Are they the same?