Here's what I am trying to do:
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
Much of the data was collected from Wikipedia and SE.
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).
- "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:
- 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!
- Is this template in Wikipedia correct? Else, are there any counter examples?
