For $I,J$ ideals $P$ Prime ideal, show that $IJ\subset P \iff I\cap J \subset P$

Question

Question : Prove the following equivalence

$IJ\subset P \iff I\cap J \subset P \iff$ $I$ or $J \subset P$

I was able to do this

$IJ \subset I$ and $IJ \subset J$ so $IJ \subset P$

$IJ \subset I$ and $IJ \subset J$ so $IJ \subset I \cap J \subset P$

Let $r \in I$ $s \in J$, so $rs \in I \cap J \subset P$ as $P$ is prime ideal so either $r \in P$ or $s \in P \implies I \subset P$ or $J \subset P$

Answer 1

You already showed that $IJ\subset I\cap J$. So one side is obvious. That is, if $I\cap J\subset P$ then $I J\subset P$.

For the other side, assume $I J\subset P$ and assume there is $x\in I\cap J$ such that $x\not \in P$. Then since $P$ is prime $x^2\not\in P$, but this is a contradiction since $x^2\in IJ\subset P$.

Answer 2

You have proven that if $I\cap J\subseteq P$ then $IJ\subseteq P$. Also if $I$ or $J$ is contained in $P$ then $I\cap J\in P$.

Now we need to prove that if $IJ\subseteq P$ then $I\subseteq P$ or $J\subseteq P$. You can prove this by contradiction.