This is the exercise.
Let $k$ be a field. Prove that $k[x]\times k[x]$ contains a principal ideal of codimension $1$, although it's not a domain.
Now, I have to find a principal ideal prime, such that it contains properly another prime. I was trying to understand how it is possible. If we consider $P=<(p(x),q(x))>$ it is never prime because we have $(p(x),1)\cdot (1,q(x))\in P$. If we consider $<(p(x),0)>$ then it's useless because it's the same that considering a principal prime ideal in $k[x]$, and we know that in $k[x]$ it doesn't exist. Then, I was thinking that in general: $P$ is prime in $A \Leftrightarrow$ $P\times A$ (and obviously $A\times P$) is prime in $A\times A$. So, how can I find thise prime? It seems impossible ;)
Hint. What about the ideal generated by $(1,x)$?