Let $A\subset B$ be a ring extension where every prime ideal of $A$ is contraction of a prime ideal in $B$. We have to find prime ideals $P_1\subset P_2$ in $A$ and a prime ideal $Q_1$ in $B$ such that $P_1=Q_1\cap A$ and there is no prime ideal $Q_2$ of $B$ with $Q_1\subset Q_2$ and $P_2=Q_2\cap A$.
If $A$ be an integral domain then $(0)$ is a prime ideal in $A$. If we can find one $A\subset B$ where lying over holds and $(0)$ of $A$ is contraction of a maximal ideal of $B$ with $A$ not being a field then we will be done. I am not being able to find one such.
