Let $R$ be a commutative ring with unity. Let $P,I,Q$ be ideals of $R$ such that $P$ and $Q$ are prime ideals and $P\subseteq I \subseteq Q$. Let $a\in R$ and $b\in R\setminus Q$. I would like to have the following property: $$ab\in I \iff a\in I.$$
Of course right to left follos from $I$ being an ideal, and the converse follows trivially when $Q=I$ or $P=I$. I could not prove it or find counter examples in the general case.
This is not true. Let $R = \mathbb{Z}[x]$, let $Q = (2, x)$, $I = (6, x)$, $P = (x)$. We know that $P, Q$ are prime ideals, and it is not difficult to see that $P \subset I \subset Q.$
Consider $2 \in R$, and $3 \in R\setminus Q$. Note that $2 \cdot 3 = 6 \in I$, but $2$ is not in $I$.