Easy counterexamples for each containment $Field\subsetneq ED \subsetneq PID \subsetneq UFD \subsetneq GCD \subsetneq Integral\space Domain$ $\mathbb Z$ 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?
 A: 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 ($\mathbb Z[\sqrt{-5}]$)
A: 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.
