vanishing set of a union
1. Proposition
Let \(A\) be a commutative ring and \(T_i \subseteq A\) a family of subsets Then
\begin{align*} V( \sum T_i) =& V(\bigcup T_i) \\ =& \bigcap V(T_i) \end{align*}2. Proof
2.1. a)
corollary of:
2.2. b)
By definition, \(\mathfrak{p} \in V(\bigcup T_i)\) if and only if \(\bigcup T_i \subseteq \mathfrak{p}\). This is equivalent to
\begin{align*} \mathfrak{p} \in& V(T_i) \forall i \in I \\ \mathfrak{p} \in& \bigcap V(T_i) \end{align*}