Is this a g.c.d. domain? Then why?

Question

Let the ring $R = \mathbb{Z}[\sqrt d]$ for $d \in \mathbb{Z}$

Then, is the domain $R$ a g.c.d. domain?

(Intuitively it looks like true, but couldn't figure out the reason of that. )

Any help would be thankful.

Answer 1

This question is already solved before.

The conclusion is the $R$ is not g.c.d. domain When we consider general case without the special cases like when the $R$ is UFD.(e.g. gaussian integer ring $Z[i]$)

Hence For the $R$, we can't say any two elements have a g.c.d.

Here is the reason.

the link https://mathoverflow.net/questions/11105/an-example-of-two-elements-without-a-greatest-common-divisor