We use the following facts about radicals without proof:

1. $r\left(\bigcap _{n=0}^k{a}_n\right)=\bigcap _{n=0}^{k}r(a_n)$
2. If $\varphi :A\to B$ is a ring morphism, then $r({\varphi }^{-1}(b))={\varphi }^{-1}(r(b))$ for any ideal $b\subseteq B$.
3. If $\mathfrak{q}\subseteq B$ is $\mathfrak{p}$-primary, then ${\varphi }^{-1}(\mathfrak{q})\subseteq A$ is 1;;${\varphi }^{-1}(\mathfrak{p})$-primary.
4. If $S\subseteq A$ is mulitplicatively closed, then the ideals of $S^{-1}A$ are exactly the ideals of $A$ which do not meet $S$.