Examples and counter-examples for rings

Question

Here's what I am trying to do:

1. Listing mnemonics used:

   • $D_n(\Bbb R)$ to denote diagonal $n \times n$ matrices with real coefficients.
   • I.D. - Integral domain, E.D. Euclidean Domain

2. Much of the data was collected from Wikipedia and SE.

3. Using this template from Wikipedia,

$$\text{Commutative rings}\supset\text{I.D.}\supset\text{U.F.D.}\supset\text{P.I.D.}\supset\text{E.D.}\supset\text{Fields}\supset\text{Finite Fields}$$

each column is a subset of previous column, in other words, examples for UFD works for Integral Domain (ID) and commutative rings, and so on (excluding non-commutative rings).

4. "Examples" given in bold are counter examples to the category to its right. That is, $\Bbb Z\times \Bbb Z$ is a Commutative Ring, but not an Integral Domain.

My doubts are:

1. Can someone verify if the data I gathered is correct and it will be great if someone can provide more simple examples in each category!
2. Is this template in Wikipedia correct? Else, are there any counter examples?

Answer 1

1. I didn't spot any mistakes so far. As for more examples, you may find some at the DaRT.

2. Yes, the template is fine.