Give an example of a UFD having a subring which is not a UFD. Give an example of a UFD having a subring which is not a UFD.
I thought of $\mathbb{Z}[\sqrt{2},\sqrt{3}]$.
Could you please explain my question. I am trying grasp the concepts, need help.

Maybe I am beating the dead horse, but...
$\mathbb{Q} + X \mathbb{R}[X] \subset \mathbb{R}[X]$
is one of many similar examples.
