I'm having some difficulties with this problem ($\mathbb{K}$ is algebraically closed).
Let $R\subseteq R'$ be an integral ring extension and let $R$ be a finitely generated algebra. Let $P_1\subsetneq P_3$ be prime ideals in $R$ and let $P_1'\subseteq P_3'$ be prime ideals in $R'$ such that $P_1'\cap R=P_1$ and $P_3'\cap R=P_3$. Prove that if there is a prime ideal $P_2\subseteq R$ such that $P_1\subsetneq P_2 \subsetneq P_3$, than there is a prime ideal $P_2'\subseteq R'$ such that $P_1'\subsetneq P_2' \subsetneq P_3'$.
My attempt By using "going up" we know that there is a $P_2''$ such that $P_1\subsetneq P_2''$ and $P_2''\cap R=P_2$. The problem now is making $P_2''$ fit into $P_3$. So, I tried to use $P_2''\subseteq P_3$ but this is not prime in general. So I kinda don't know what to do...