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Is there a 'Galois theory' with fields replaced by (non-commutative) division rings?

I have googled this, and it seems that there are known results in that direction, for example, this paper which says: "Galois theory was extended to division rings by N. Jacobson and H. Cartan, and to more general rings by G. Azumaya and T. Nakayama", or this paper which says: "The author has formulated sufficient conditions on a ring $B$ and a group $G$ of automorphisms of $B$ to derive a Galois theory of noncommutative rings which extends the Galois theory of commutative rings developed by Chase, Harrison, and Rosenberg".

Can one please explain the main ideas and results known or give a reference for a book on the subject?

Thank you very much!

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I’m not sure that this book contains exactly what you want,but “Galois Theories” by Bordeaux and Janelidze contains “Galois theory for commutative rings”.It also contains “pure categorical” Galois theory, which is effective for pure categorical situations.(Actually, Galois theory for commutative rings is one of the corollaries of categorical Galois theory)

I hope it will help you.

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  • $\begingroup$ Thank you. Sounds interesting. $\endgroup$ – user237522 Jul 5 at 14:44

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