Easy counterexamples for each containment $Field\subsetneq ED \subsetneq PID \subsetneq UFD \subsetneq GCD \subsetneq Integral\space Domain$

Question

$\mathbb Z$ could be used for an example of an Euclidean Domain that is not a Field. Can you give other easy examples for the other proper inclusions?

Answer 1

Yeah, I wrote a website to do that. I'll also annotate with a concrete example in case the site goes down.

Euclidean domain, not a field ($k[x]$ for a field $k$.)

Principal ideal domain not a Euclidean domain ($\mathbb Z[\frac{1+\sqrt{-19}}{2}]$)

Unique factorization domain, not a principal ideal domain ($\mathbb Z[x]$)

Domain, not a unique factorization domain ($\mathbb Z[\sqrt{-5}]$)

Answer 2

Here are key examples, where $K$ is any field:

$$\bbox[5px,border:2px solid black]{ \bbox[5px,border:2px solid black]{ \bbox[5px,border:2px solid black]{ \bbox[5px,border:2px solid black]{ \begin{eqnarray} & \text{Euclidean domain} \\ & \mathbb{Z}\ \ \ \mathbb{Z}[i]\ \ K[x] \\ \end{eqnarray} } \begin{eqnarray} & \text{PID} \\ & \mathbb{Z}\left[\frac{1 + \sqrt{19}i}{2}\right] \\ \end{eqnarray} } \begin{eqnarray} & \text{UFD} \\ & \mathbb{Z}[x]\ \ \mathbb{Z}[x, y]\ \ K[x, y] \\ \end{eqnarray} } \begin{eqnarray} & \text{ integral domain } \\ & \mathbb{Z}[\sqrt{5}i] \\ \end{eqnarray} }$$

For the PID which is not a Euclidean domain, there does not seem to be an easy example, compared to the other examples listed.