Proposition 17

Let $A$ be a ring, $M$ an $A$-module and $a$ its annihilator; if $M$ is finitely generated, then $\operatorname{Supp}(M) = V(a)$ and $\operatorname{Supp}(M)$ is therefore closed in $\operatorname{Spec}(A)$. Let $(m_i)_{1 \leq i \leq n}$ be a system of generators of $M$ and let $a$, be the annihilator of $m_i$; then $a = \bigcap_{i=1}^n a_i$, hence $V(a) = \bigcup_{i=1}^n V(a_i)$ (no. 3, equation (3)) and the Proposition follows from the Corollary to Proposition 16.