vanishing set of an intersection of ideals

Let \(\mathfrak{p} \in V(\mathfrak{a}_1) \cup V(\mathfrak{a}_2)\), then w.l.o.g. \(\mathfrak{a}_1 \subseteq \mathfrak{p}\). Therefore \(\mathfrak{a}_1 \cap \mathfrak{a}_2 \subseteq \mathfrak{p}\) and \(\mathfrak{p} \in V(\mathfrak{a}_1 \cap \mathfrak{a}_2)\) (see: Vanishing set and order reversing)

Since \(\mathfrak{a}_1 \cap \mathfrak{a}_2 \supseteq \mathfrak{a}_1 \cdot \mathfrak{a}_2\), we get by the order reversing property of the vanishing set

\begin{align*} V(\mathfrak{a}_1 \cap \mathfrak{a}_2) \subseteq V(\mathfrak{a}_1 \cdot \mathfrak{a}_2) \end{align*}

Let \(\mathfrak{p} \in V(\mathfrak{a}_1 \cdot \mathfrak{a}_2)\), then \(\mathfrak{a}_1 \cdot \mathfrak{a}_2 \subseteq \mathfrak{p}\) and hence w.l.o.g. \(\mathfrak{a}_1 \subseteq \mathfrak{p}\) (see: prime ideal and product). Then we get

\begin{align*} \mathfrak{p} \in V(\mathfrak{a}_1) \end{align*}

and therefore

\begin{align*} V(\mathfrak{a}_1 \cdot \mathfrak{a}_2) \subseteq V(\mathfrak{a}_1 ) \cup V(\mathfrak{a}_2) \end{align*}