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.

Take any integral domain which is not a UFD and consider it as a subring of its field of fractions. (Fields are UFD for trivial reasons and if you don't accept this, take the polynomial ring over it)
