If $\displaystyle A,B$ are principal ideal rings, then $\displaystyle A \times B$ is also principal ideal ring.
Is this right? Why??
yes, if $\displaystyle A$ and $\displaystyle B$ both have $\displaystyle 1$ of course. the reason is that every ideal of $\displaystyle A \times B$ is in the form $\displaystyle I \times J$ for some ideal $\displaystyle I$ of $\displaystyle A$ and some ideal $\displaystyle J$ of $\displaystyle B$.
maybe i don't have to mention that $\displaystyle A \times B$ is never a domain. so the result doesn't hold for principal ideal "domains".