vanishing set, generated ideal and intersection
1. Proposition
Let \(A\) be a commutative ring, \(T \subseteq A\), \(I = (T)\) the generated ideal. Then
\begin{align*} V(T) =& V(I) \\ =& V(\bigcap_{\mathfrak{p} \in V(T)} \mathfrak{p}) \end{align*}2. Proof
2.1. 1) \(\supseteq\) 2)
follows from \(T \subseteq I\)
2.2. 2) \(\supseteq\) 3)
follows from \(I \subseteq \bigcap \mathfrak{p}\)
2.3. 3) \(\supseteq\) 1)
Let \(\mathfrak{p} \in V(T)\), then \(\mathfrak{p} \in V(\mathfrak{p}) \supseteq V(\bigcup \mathfrak{p}\) and therefore also
\begin{align*} \mathfrak{p} \in V(\bigcap_{\mathfrak{p} \in V(T)} \mathfrak{p}) \end{align*}