Corollary

If the ring A is Noetherian, the mapping $\mathfrak{p} \mapsto S^{-1}\mathfrak{p}$ is a bijection of $\mathrm{Ass}_A(M) \cap \Phi$ onto $\mathrm{Ass}_{S^{-1}A}(S^{-1}M)$.

If A is not Noetherian, the mapping $p \mapsto S^{-1}p$ of $\mathrm{Ass}_A(M) \cap \Phi$ to $\mathrm{Ass}_{S^{-1}A}(S^{-1}M)$ is not necessarily surjective (Exercise 1).