Let $A$ be a ring, $S\subseteq A$ multiplicatively closed. Let $\mathfrak{p}\in Spec(A)$ such that $\mathfrak{p}\bigcap S=\mathrm{\varnothing }$. Then:

1. If $\mathfrak{a}\subseteq A$ is a $\mathfrak{p}$-primary ideal of $A$ such that $\mathfrak{a}\bigcap S=\mathrm{\varnothing }$, then 1;; $S^{-1}\mathfrak{a}$ is an $S^{-1}\mathfrak{p}$-primary ideal of $S^{-1}A$, and $(S^{-1}\mathfrak{a}{)}^c=\mathfrak{a}$.
2. If $\mathfrak{b}\subseteq B$ is an $S^{-1}\mathfrak{p}$-primary ideal of $S^{-1}A$, then 1;;${\mathfrak{b}}^{\mathfrak{c}}$ is a $\mathfrak{p}$-primary ideal of $A$.
3. There is a bijection:

PROOF:

• Use that, for localization, ${\mathfrak{a}}^{ec}=\bigcup _{s\in S}(\mathfrak{a}:s)$.