Prove the following:
Let $I_1$ and $I_2$ be ideals of a ring $R$ such that $I_1 \not\subseteq I_2$ and $I_2\not\subseteq I_1$. Then $I_1\cap I_2$ cannot be a prime ideal.
Take $\Bbb{Z}[x,y,z]$. Clearly $(x,y)\not\subseteq (x,z)$ and $(x,z)\not\subseteq (x,y)$. However, $(x,y)\cap(x,z)=(x)$ is a prime ideal in $\Bbb{Z}[x,y,z]$. Isn't this a contradiction?
If $a \in I_1 \backslash I_2$ and $b \in I_2 \backslash I_1$ then $$ab \in I_1 \cap I_2 \, \mbox{ but } \, a,b \notin I_1 \cap I_2$$