Let $\mathfrak{a}\subseteq A$ be an ideal of a ring, and $S\subseteq A$ be multiplicatively closed. Suppose that $\mathfrak{a}={\mathfrak{q}}_1\bigcap \dots \bigcap {\mathfrak{q}}_n$ is a (minimal) primary decomposition of $\mathfrak{a}$. Without loss of generality suppose that $r({\mathfrak{q}}_{\mathfrak{i}})\bigcap S={\mathfrak{p}}_i\bigcap S=\mathrm{\varnothing }$ for $i\le m\le n$, and $r({\mathfrak{q}}_i)\bigcap S={\mathfrak{p}}_i\bigcap S\ne \mathrm{\varnothing }$ for $m<i\le n$. Then:

1. $S^{-1}\mathfrak{a}=S^{-1}{\mathfrak{q}}_1\bigcap \dots \bigcap S^{-1}{\mathfrak{q}}_{\mathfrak{m}}$
2. $(S^{-1}\mathfrak{a}{)}^c={\mathfrak{q}}_{\mathfrak{1}}\bigcap \dots \bigcap {\mathfrak{q}}_m$
   Are both 1;; minimal primary decompositions in $S^{-1}A$ and $A$ respectively.