TITLE: Give an example of a UFD having a subring which is not a UFD.
QUESTION [13 upvotes]: 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.

REPLY [6 votes]: $\mathbb{Z}[2i]$ as a subset of $\mathbb{Z}[i]$. First one is not integrally closed, so can't be UFD, second is Euclidean.