I am studying for an upcoming exam, and I came across the following statement that I am struggling to prove:
If $P$ is a prime ideal, then $P$ cannot be the intersection of two ideals that properly contain $P$.
So far, I have: If $P$ is properly contained in ideals $I$ and $J$, then we know that $P \subset I\cap J$. So I want to show that $ I\cap J $ is not in $P$. Also, $P$ is prime $\implies$ if $ab\in P$ then at least $a \in P$ or $b \in P$. So, I have been trying to show that there is an element $x \in I \cap J$ such that $x$ is not in $P$, but I keep getting stuck at this step. Any help would be greatly appreciated!