# 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?

• $\mathbb{Z}[X,Y]$ is a UFD which is not a PID. – Hayden Jun 3 at 14:10
• $k\subset \mathbb Z \subset \mathbb Z[1+\sqrt{-19}/2] \subset \mathbb Z[x] \subset \mathbb Z[\sqrt{-5}]$ – Mustafa Jun 5 at 1:43
• Even every integral domain is contained in its field of fractions. – Kumar Jun 5 at 12:03

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 ($$\mathbb Z[\sqrt{-5}]$$)

• Cool website! A topological version is topology.jdabbs.com – Hayden Jun 3 at 14:17
• @Hayden Sure, I've known about them back when it used to be Spacebook. It's also linked on the front page. if you have any ideas to contribute, please check out the contribute link :) – rschwieb Jun 3 at 14:18
• Nice website! Wish it had proofs though. But I guess that's too much work for 1 person. – SKYejin Jun 3 at 14:23
• @nilpo10 Yes, it is. The best I can do in each case is to provide citations for each proposition used to deduce a particular property. If you really need to know something, you can contact me through the site and I can find the chain of inference. Actually, most of the examples I gave above have proofs on the site here. – rschwieb Jun 3 at 14:25