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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.

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    $\begingroup$ group rings${}$? $\endgroup$ – Lord Shark the Unknown Apr 23 '18 at 16:48
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    $\begingroup$ Extending the example suggested by @LordSharktheUnknown there are also symmetric algebras, which have many of the same properties a group algebras. $\endgroup$ – Tobias Kildetoft Apr 23 '18 at 17:28
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My site (Database of ring theory) presently is at 90+ examples (and growing) and certainly contains many rings at all levels that might interest you. So I'd basically like to suggest any/all of those examples as starters.

Things as simple as the integers all the way up to some really exotic rings like O'Meara's infinite matrix algebra (which incidentally contains a few complex examples by Bergman) are all there. You can search by properties or by construction keyword to get, for example, triangular rings. That is perhaps how you might explore some of the possibilities of constructions.

Here is the comprehensive list of properties currently in the database.

The meters on the browse page indicate how much the database knows about them, but that is not a complete picture of how much is known, of course. If you happen to find out something that is listed as unknown, or find an interesting ring or property you'd like to suggest including, please drop me a line.

The ring for which I know the least is this relatively new example due to Šter.

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  • $\begingroup$ You may or may not also be interested in this earlier question where someone was asking about a general purpose reference for ring examples. $\endgroup$ – rschwieb Apr 23 '18 at 19:41

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