# Examples of interesting rings to study during an undergraduate course in non-commutative rings.

I'm taking a course on Modern Algebra at my university and my professor keeps asking us to look for examples of rings that may be interesting to discuss in class. For instance, he dedicated some time today to discuss upper triangular rings which has been discussed a lot here. See for instance here. He discussed the submodules of this ring $A = \begin{bmatrix} R & M \\ 0 & S \\ \end{bmatrix}$, writing them as tuples $(M_1,M_2,\phi)$ where $M_1$ is a right $R$-module, $M_2$ is a right $S$-module and $\phi:M_1 \times M \to M_2$ is bilinear. So my question is, can you propose any interesting rings assuming the contents that I leave indicated below? Any ring that you specially like? Maybe a ring that has not been fully studied?

1. Brief history of Modern and Abstract Algebra and Category Theory.
2. Rings and ring constructions. Artinian and noetherian rings. Ring decompositions. Idempotent elements. Prime rings and ideals. Radical is prime. Semiprime artinian rings. Artin-Wedderburn theorem. Right primitive rings. Jacobson density theorem. Jacobson radical. Essential extensions and inyective hulls. Application to noetherian rings.
3. Modules and non-commutative rings representations. Bilateral modules, tensor product. Lattice of submodules. Finitely generated modules. Artinian and noetherian modules. Free modules. Semisimple modules. Generation and cogeneration. Generators and cogenerators in modules. Projective and injective modules. Indecomposable modules.
4. Categories. Functors. Direct sum. Direct product. Equivalencies of modules categories. Characterization of equivalences between module categories. Equivalent rings.

Please note I'm using the soft-question tag and feel free to suggest any improvement in the above.

• group rings${}$? – Lord Shark the Unknown Apr 23 '18 at 16:48
• Extending the example suggested by @LordSharktheUnknown there are also symmetric algebras, which have many of the same properties a group algebras. – Tobias Kildetoft Apr 23 '18 at 17:28