Proposition

Show that the intersection of any collection of prime ideals is radical.

Solution

Let $\{ I_{\alpha} \}$ be a collection of prime ideals. Let $f \in k[x_1, \cdots, x_n]$. Suppose $f^m \in \bigcap_{\alpha} I_{\alpha}$ for some $m \geq 1$. For all $\alpha$, $f^m \in I_{\alpha}$ implies that $f \in I_{\alpha}$. Therefore, $f \in \bigcap_{\alpha} I_{\alpha}$, so the intersection of any collection of prime ideals is radical.