When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand. 


There are many convenient examples of commutative rings, both finite and infinite, and there are many convenient examples of infinite non-commutative rings. But I don't have a good collection of finite non-commutative rings to think about. I usually just think of a matrix ring over a finite field.


Do YOU have other examples that you particularly like/find easy to use/find to be a good source of counterexamples?



What are your favorite finite non-commutative rings?
2 families of examples that are sometimes useful to have in mind:


(1) The group ring of a non-abelian finite group over a finite commutative ring.


and 


(2) the incidence algebra of a finite poset over a finite commutative ring (the ring of upper triangular matrices is a basic example of this). 


Of course, both of these are special cases of the same more general categorical (or quiver) definition. Before I wrote that I'd never dealt with the more general concept, but that was a lie...


