I'm currently reading "Introduction to Ring Theory" by Paul Cohn
As I'm working through it, I'm realizing I'm really deficient in having examples of various properties at the ready, and it makes it hard to establish a mental picture of what's happening as I read - for instance, a theorem may have the hypotheses "Let $R$ be a simple ring" or "Let $R$ be an Artinian ring" - and I have difficulty conjuring examples to use as I try to understand what the theorem is saying. This is especially difficult because most of the rings that come to mind are fields or at least commutative, and these cases tend to trivialize many of the theorems in the book.
How does one start building up a collection of rings with various properties to use as examples when those properties are invoked? Are there references that exist for this already?
EDIT: I've decided to include some of my background - I'm an undergraduate student (nearly graduated) in math, with a relatively heavy algebra background - I've taken 4 semesters of algebra classes, two of which were at the graduate level. I'm familiar with introductory modern algebra, I'm just lacking in the "canonical example of structure X with property P" department.
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.