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Can you please provide references how non-communative ring theory works in mathematics outside non-commutative ring theory?

I am interested in applications in the following fields: topology, geometry, algebraic geometry, invariant theory, algebraic number theory, combinatorics, combinatorial geometry, convex polyhedra, K-theory, Lie theory, PDE.

I am especially interested in those (concrete) applications that allows:

  1. compute something

  2. classify something

  3. prove existence of something

I do not expect detailed answers, this is just reference request. However, you are welcome to provide deep answers that I will understand in several years from now. Concrete computations, books and reviews are highly appreciated.

Thanks for your time!

Update

Please continue to provide answers, I will be happy to upvote every relevant answer!

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    $\begingroup$ Depends on what you mean by non-commutative. Cohomology theories often give rings which are non-commutative graded rings (which are often called "commutative" because, well,...). $\endgroup$
    – Randall
    Oct 25, 2017 at 11:49
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    $\begingroup$ In quantum mechanics, measurable quantities (observables) are represented as members of a ring of operators on the set of wave functions. Irwin Schrodinger found that the Position operator and the Momentum operator did not commute, which is the heart of the Uncertainty Principle. $\endgroup$ Oct 25, 2017 at 15:24
  • $\begingroup$ Matrix rings,,,...... $\endgroup$ Nov 3, 2018 at 3:55

2 Answers 2

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You may want to have a look at these:

"Ideals over a Non-Commutative Ring and their Application in Cryptology"

"Noncommutative Rings and Their Applications"

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    $\begingroup$ Thanks a lot for your answer, I will try to check this references. $\endgroup$
    – Hedgehog
    Oct 25, 2017 at 11:58
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I would say that applications of Clifford algebras (also see "geometric algebra") would interest you.

Ring theory explains most of the structure and property of such algebras, but it seems like there has also been a lot of excitement in the past few decades about applying them both to geometry, Lie algebra, physics, computer science, computer vision, and other computational geometry-type problems.

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  • $\begingroup$ Wow! I new several basic definitions about Clifford algebras, but I had no idea that noncommutative rings are important there. Thanks a lot for the reference. $\endgroup$
    – Hedgehog
    Oct 25, 2017 at 13:18
  • $\begingroup$ @Hedgehog Well sure... the only two commutative real Clifford algebras are $\mathbb R$ and $\mathbb C$, and the rest (infinitely many) are noncommutative. $\endgroup$
    – rschwieb
    Oct 25, 2017 at 16:02

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