Applications of non-commutative ring theory in mathematics (outside non-commutative ring theory)

Question

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!

Answer 1

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.

Answer 2

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"