I am taking a graduate course in noncommutative algebra at a university that is not my home university. Although we had some basic noncommutative algebra at my home university, the focus was always on the commutative cases.
The fact that I often have to look up theorems and proofs about noncommutative rings which I never learned (in the noncommutative case) is a bit of an annoyance but is something I can mostly overcome. What really hinders me is my lack of knowledge of good examples of noncommutative rings. I can mostly do exercises which require me to prove a given claim, but not exercises which require me to come up with counterexamples (e.g. 'find a ring which is simple but not left noetherian', 'find a left noetherian ring with a nil ideal that is not nilpotent', ...).
The only class of noncommutative rings I understand fairly well are matrix rings over a certain ring. Because the ideal structure of these rings is so closely related to that of their base ring, matrix rings often cannot serve as counterexamples. I know that one can generalize matrix rings to endomorphim rings over arbitrary modules and that there exists something like 'noncommutative polynomial rings', where $XY$ and $YX$ are different objects. But I really do not grasp these rings. When are they (left, right) noetherian/artinian, when are they (semi)simple? What are their ideals, in particular their Jacobson radical, prime ideals, ... ?
I feel like I will be losing too much time if I have to look all of this up individually. Is there some encyclopedia of noncommutative rings, or a crash course in examples of noncommutative rings and their properties?