Proposition 20

Let $A$ be a ring and $M$ a finitely generated $A$-module. For every prime ideal $p \in \operatorname{Supp}(M)$, there exists a non-zero $A$-homomorphism $w : M \to A/p$.

Let $p \in \operatorname{Supp}(M)$. As $M$ is finitely generated and $M_p \neq 0$,

(§ 3, no. 2, Proposition 4). Let $K = A_p/pA_p$ be the field of fractions of the integral domain $A/p$; since $M_p/pM_p$ is a vector space over $K$, which is not reduced to 0, there exists a non-zero linear form $u : M_p/pM_p \to K$. If $(x_i)_1 <_n^*$ is a system of generators of $M$, $\bar{x}_i$ the image of $x_i$, in the $(A/p)$-module $M_p/pM_p$, there exists an element $\alpha \neq 0$ of $A/p$ such that the $\alpha u(\bar{x}_i)$ belong to $A/p$ for $1 \leq i \leq n$; hence $v = \alpha u$ is a non-zero $(A/p)$-linear mapping from $M_p/pM_p$ to $A/p$. The composition is therefore the required homomorphism.