Counter Ring

Counter Ring is the (temporary) name of a community-based project, whose purpose is to collect, maintain, and even construct counterexamples in the theory of rings.

What is this project about

Hundreds of properties of rings were studied within Ring Theory. Sometimes it is known that every ring with properties A and B has property C as well; sometimes the problem is still open; and very often, it was settled long ago by a clever counterexample which is only known to a handful of experts.

The main goal of this project is to collect non-trivial examples and present them in way that is reasonably easy to search.

How do I join?

Contributing is simple:

1. To create an account, go to the main page, and log-in through a gmail account (is it currently impossible to choose your login-name; it will be set-up by the interaction of this website with gmail).
2. To log-in after you have an account, press the link at the upper-left corner
   1. (login = "כניסה לחשבון").
3. Once you have an account, you can change the interface language by going to Preferences; the second section in the first tab.
   1. (Preferences = "העדפות").
4. Start working
   1. (edit = "עריכה")

The website has some (though limited) TeX support. The code <math>M_n(F[x])</math> will display [math]\displaystyle{ M_n(F[x]) }[/math], etc.

About the website

The project runs on a Wiki-like platform, currently hosted on "math-wiki", which was created by Erez Scheiner, mostly for the students at the mathematics department at Bar Ilan University. Hence the Hebrew user interface and some awkward left-to-right issues.

At some point we'll move to our own server.

What goes where

The project has two parts:

• Examples. Examples should be precisely described. If the details are not easy to verify, please include a reference to the literature. The material will be organized into chapters and sections as it accumulates.
• Queries. Is every finitely generated nil algebra necessarily nilpotent? If there are examples you think the project should cover, list them here.

General comments can be posed in the talk page associated with this page ("שיחה").

For technical questions, email Uzi Vishne (vishne at math.biu.ac.il).

Examples

• A ring which is not commutative: [math]\displaystyle{ M_n(\mathbb{Q}) }[/math].
• An infinite ring, all of whose quotients are finite: [math]\displaystyle{ \mathbb{Z} }[/math].
• A Euclidean integral domain, quadratic over the integers, whose standard norm is not Euclidean: [math]\displaystyle{ {\mathbb{Z}}[\sqrt{14}] }[/math].