#noncommutative algebra

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Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

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Would people be interested in a video course on noncommutative algebra? The prerequisite knowledge would be linear algebra, basic group module theory and in the end some knowledge of Galois theory. Topics I might teach would include: Jacobson Radical Theory Central Simple Algebras Wedderburn-Artin and Wedderburn-Mal'tsev The Double Centraliser Theorem Noether-Skolem Theorem Brauer group Theory Crossed Products and Galois Theory Group Cohomology, briefly, and connection with the Brauer Group Further topics to be decided (Maybe Morita Theory, or more Brauer Theory).

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I'm writing notes for the youtube course and I just remembered that there are rings for which the categories of left R-Modules and right R-Modules are not equivalent. This is extremely cursed.

You don't get to see this stuff in commutative algebra!

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Just finished the second episode in my noncommutative algebra series!

Please check it out!

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Hi please listen to me rant about some cool algebra/geometry I saw this week (Field Patching, Noncommutative Geometry)