I am trying to solve problem 11 in Dummit and Foote section 7.4. The problem is the following:
Assume $R$ is commutative. Let $I$ and $J$ be ideals of $R$ and assume $P$ is a prime ideal of $R$ that contains $IJ$. Prove either $I$ or $J$ is contained in $P$.
I came up with the following proof and I just want to check if it is right. It's as follows:
We know $IJ \subset P$. Then if we consider $i \in I$ and $j \in J$. Then, we know $ij \in IJ \subset P$. By primality of $P$ and since $ij \in P$, we know that either $i \in P$ or $j \in P$. Thus, we must have that $I \subset P$ or $J \subset P$, as desired.
I would appreciate any suggestion or comments on this proof. Thanks!