Existence of an element in a ring with a given property

Question

Let $R$ be a ring (commutative, with unity), $\mathfrak p\subset R$ a prime ideal, $M$ a finitely generated $R$-module such that $S_1^{-1}M=0$, where $S_1=R\setminus \mathfrak p $. How does one show that there is an $r∈R$ such that the set of all prime ideals in $R$ not containing the ideal $(r)$ has $\mathfrak p$ as one of its elements and possesses the property that for any other $\mathfrak q$ from this set, $S_2^{-1}M=0$, where $S_2=R\setminus \mathfrak q$?