Chain of inclusions between classes of rings I need to show using a chain of inclusions the relationship between the types of rings which are = {rings, commutative rings, integral domains, Euclidean domains, principal ideal domains, unique factorization domains, fields} 
I know that {rings}\supset {commutative rings}\supset ???
Any advice on how to proceed and finish this chain?
 A: 
fields $\subsetneq$ Euclidean domains $\subsetneq$ principal ideal domains $\subsetneq$ unique factorization domains $\subsetneq$ integral domains  $\subsetneq$ commutative rings $\subsetneq$ rings.

Some of these strict inclusions are easy to see, some not. For those, you need theorems that you can find in standard books in ring theory (like the book "Algebra" of "Hungerford").
To show that inclusions are strict, you should find examples. Some of these examples are not easy to find, or are hard to prove that satisfy the conditions (for beginners). e.g. $k[x,y]$ is unique factorization domain but not principal ideal domain. So the inclusion $\text{principal ideal domains} \subseteq \text{unique factorization}$ domains is strict.
A: Given your question, I assume that the word 'field' means commutative field. In many contexts it does.
Said that, you should proceed this way: pick two classes and see if every element of the first is of the second, or the other way around. For example: is every Euclidean ring a commutative ring? The answer is yes, so the class of commutative rings contains the class of Euclidean rings.
If your teacher has assigned this problem, surely you have the theorems to complete the chain.
