Product of two principal ideals in $\Bbb Z[x]$

Question

I'm looking for an easy argument for the following question:

True or false, and why: The product of two principal ideals in $\Bbb Z[x]$ is a principal ideal.

I know that $\Bbb Z[x]$ is not a PID, so it doesn't need to be true. How can I show it is or what is a counterexample?

Answer 1

It is true. In fact, it holds in every commutative ring (with $1$):

The product of two principal ideals is a principal ideal.

The product $IJ$ of two ideals $I$ and $J$ is the smallest ideal containing all the products $ij$ with $i\in I$ and $j\in J$.

If $I$ and $J$ are principal, then $I=aR$, $J=bR$ and so $IJ$ is the smallest ideal containing all elements of the form $(ar)(bs)=(ab)(rs)$, with $r,s \in R$. This is exactly the principal ideal $abR$.