Example

Let $a$ be an ideal of $A$; in the notation of no. 3, we have

If $p$ is a prime of $A$ such that $a \notin p$, then $(A/a)_p = 0$ (§ 3, no. 1, Remark 3); if on the other hand $a \subset p$, $aA_p$ is contained in the maximal ideal $pA_p$ of $A$, and $(A/a)_p$ is isomorphic to $A_p/aA_p$ and hence is non-zero (§ 3, no. 1, Proposition 3); whence our assertion. In particular, $\operatorname{Supp}(A) = \operatorname{Spec}(A)$.