In any discipline, examples are indispensable tools. They teach us the limits of our knowledge by showing us what is possible, and they can often illuminate the differences between concepts.
Here is the problem: In the past, students and researchers would incrementally accumulate a stockpile of examples over a lifetime. Quite often each example is remembered with a specific purpose, and other properties are perhaps unobserved. Faster access to examples was made possible by books like Steen and Seebach's Counterexamples in topology, but still, retrieval is limited a bit by manual use of the book. (Update: I recently was introduced to Hutchins' Examples of commutative rings. 1981, which is now the largest text collection of ring examples -- all commutative-- that I'm aware of.)
The DaRT, leveraging the power of software, can do much more. Not only can we maintain an ever-growing store of examples, we can also efficiently explore them, and we can automate the process of "completing the picture" for each example. We can also smoothly track multiple citations for items in a way that books would have a hard time doing. In all, DaRT can provide a more complete, more easily searchable, and a better referenced package than a book.
For convenience, you can explore the set of examples in two ways. You can explore the full database, or if you are only interested in commutative rings and their properties, you can search the database through the lens of the commutative version of browsing and searching. So if you don't see a property of commutative rings in the full database, check the specialized commutative ring version: you will probably find it there.
In addition to rings and their properties, you can also find theorems of ring theory, many of which are fuel for the automated deduction scripts.
Have fun exploring the data, and be sure to register and contribute!
This application is on GitHub! If you'd like to help make this site better, please drop me a line there inquiring about how to contribute.
I started development of this project in mid 2013 after finding out about Austin Mohr's site on topological spaces (then called "Spacebook," now reincarnated as π-Base). Thanks to Austin's helpful advice (and a bit of a peek under the hood of the site) I began dreaming of a site for rings that could track ring examples, their properties, and have some logic on the back end to fill things out.
The next major round of thanks goes to Derik Rhodes, who set me up with a development environment and sketch database in late 2013. I went on developing locally for a year while I mustered the energy to learn about how and where I could make the page live.
This finally happened when I learned about James Dabbs' work with Austin developing π-Base. Thanks to James' generous advice and encouragement, this Database of Ring Theory is now finally able to make its debut on Heroku. Thanks again for helping make this site a reality!
Ryan C. Schwiebert, December 2014