Proposition 12

For all $f \in A$, the open set $X_f$ in $X = \mathrm{Spec}(A)$ is quasi-compact; in particular, the space $X$ is quasi-compact.

As the $X_g$ form a base of the topology, it is sufficient to prove that, if $(g_\lambda)_{\lambda \in L}$ is a family of elements of $A$ such that $X_f \subset \bigcup_{\lambda \in L} X_{g_\lambda}$, then there exists a finite subfamily $(g_{\lambda_i})_{i \in I}$ such that $X_f = \bigcup_{i \in I} X_{g_{\lambda_i}}$. But the relation $X_f = \bigcup_{\lambda \in L} X_{g_\lambda}$ means that there exists an integer $n > 0$ and a finite subfamily $(g_{\lambda_i})_{i \in I}$ such that "f" belongs to the ideal generated by that subfamily (Corollary 3 to Proposition 11); whence the proposition.