A reference for ring examples

Solution 1:

Usually you just accumulate examples as you go. You find them while reading, while learning from on teachers , and gradually gain experience finding your own.

Since I also sought to learn lots of examples, I began a website, the Database of ring theory to facilitate this.

The name aspires to more, but for now its strength is as a database of examples of rings with identity.

A great way to cement an example in memory is to tie it with some theorem, or surprising property it has. For example, the endomorphism ring of a countable infinite dimensional vector space contains elements such that $ab=1$ and $ba\neq1$, and it is isomorphic to all square matrix rings over itself.