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?


Is there some encyclopedia of noncommutative rings, or a crash course in examples of noncommutative rings and their properties?

Er, well, you couldn't go wrong by just picking a book on the topic. I would recommend Lam's First course in noncommutative rings for both readability and breadth of exposure.

For a source with easy access to examples (but short on exposition, maybe) you could try my fledgeling website, the Database of Ring Theory. There are many rings which aren't commutative already in the database, and some theorem statements that you might find useful, and I welcome additional contributions.

As for getting started with good examples of noncommutative rings, matrix rings and monoid rings over nonabelian monoids will take you a long way. Another handy construction is the 'triangular ring' construction where you can build a ring out of two rings $R,S$ and an $R,S$ bimodule $M$. I'm sure you'd encounter these and more perusing what's on file at the site. Let me know if I can add additional explanations here for anything you're curious about there.

When are [matrix rings, polynomial rings over noncommuting indeterminates] (left, right) noetherian/artinian, when are they (semi)simple? What are their ideals, in particular their Jacobson radical, prime ideals, ... ?

A matrix ring is right Noetherian iff its base ring is right Noetherian. It's semisimple iff its base ring is semisimple. The Jacobson radical of a matrix ring corresponds to the set of matrices with entries from the Jacobson radical of the base ring. Similarly the prime ideals correspond to those in the base ring.

Polynomial rings with noncommuting indeterminates over commutative rings (better known as free algebras) can be quite exotic depending on the coefficient ring you supply and the number of variables. I would not expect to find as many convenient answers for this type of construction. I think you can still say that if the coefficient ring is Noetherian and there are only finitely many variables, you have a Noetherian algebra.

The big thing to watch out for in noncommutative polynomial rings, or polynomial rings with noncommuting coefficients, for that matter, is that evaluation no longer works like you want it to.

Two more things about making a crossing from commutative theory to noncommutative theory. Firstly, localization is not always possible in the same way as in commutative rings. Secondly, there is an important concept for noncommutative rings called Morita equivalence which is "invisible" in the theory of commutative rings (since Morita equivalence between commutative rings amounts to isomorphism.) Those would be two major differences to get your head around as you study.

I hope this helps you get started. As long as you have a healthy sense of skepticism while recalling your commutative knowledge in noncommutative contexts, I think you'll be in no danger.

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    $\begingroup$ +1 for the recommendation of Lam's book, which is an excellent intro to noncommutative rings; and for the quick overview of some special cases of interest as well as those that require some caution. $\endgroup$ – Chris Leary Nov 8 '16 at 16:52
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    $\begingroup$ Thanks for the exposition. Indeed, it is sometimes hard to know what intuition I can carry over to the noncommutative case. As I mentioned, I find this particularly hard when trying to find counterexamples for theorems which hold in the commutative case, but not in the noncommutative case. I'll see if I can use your tips to find some needed solutions to exercises. $\endgroup$ – Bib-lost Nov 8 '16 at 19:24

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