Remark (1)

Let $M$ be a finitely generated module over a Noetherian ring $A$; let $(M_1), \ldots$ be a composition series of $M$ such that, for $0 \leq i \leq n - 1$, $M_i / M_{i+1}$ is isomorphic to $A/p_i$, where $p_i$ is a prime ideal of $A$ ($§ 1$, no. 4, Theorem 1). If $p$ is a minimal element of $\operatorname{Ass}(M)$, the length $\operatorname{long}_{A_p}(M_p)$ is equal to the number of indices $i$ such that $p_i = p$. For the $(M_i)_p$ form a composition series of $M$, and $(M_i)_p / (M_{i+1})_p$ is isomorphic to $(A/p_i)_p$ and hence to $\{0\}$ if $p_i \neq p$ (since $p$ is minimal in the set of $p_i$ by $§ 1$, no. 4, Theorem 2) and to $(A/p)_p$ which is a field, if $p_i = p$.