Proposition 16

Let $A$ be a ring and $M$ an $A$-module. (i) If $N$ is a submodule of $M$, then

(ii) If $M$ is the sum of a family $(N_i)_{i \in I}$ of submodules, then (i) From the exact sequence $0 \to N \to M \to M/N \to 0$, we derive, for every prime ideal $p$ of $A$, the exact sequence (§ 2, no. 4, Theorem 1). For $M$, to be reduced to 0, it is necessary and sufficient that $N_p$ and $(M/N)_p$ be so. In other words, the relation $p \notin \operatorname{Supp}(M)$ is equivalent to "p $\notin \operatorname{Supp}(N)$ and $p \notin \operatorname{Supp}(M/N)$", which proves (i). (ii) For every prime ideal $p$ of $A$, $M$, is the sum of the family of submodules $(N_i)_p$ (§ 2, no. 4). To say that $M, \neq 0$ means that there exists $i \in I$ such that $(N_i)_p \neq 0$, whence (ii).