Would you like to be the contributor for the 100th ring on the Database of Ring Theory? Go here!
You want to find rings having some properties but not having other properties? Go there: Database of Ring Theory! A great repository of rings, their properties, and more ring theory stuff.
This page provides counterexamples on Clifford algebras, spinors, spin groups and the exterior algebra. You’ll find a list of counterexamples references at the end of the article.
Look at pi-Base for a gold mine of topological examples.
Want to know more about counterexample definition? Look at Wikipedia.