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What are some examples of a non commutative division ring other than quaternions?

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  • $\begingroup$ researchgate.net/post/… $\endgroup$ – user89712 Oct 28 '13 at 10:25
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    $\begingroup$ The endomorphism ring of a simple $R$-module is always a division ring. I would expect them to be in general non-commutative. $\endgroup$ – Pavel Čoupek Oct 28 '13 at 10:28
  • $\begingroup$ The Wikipedia entry for division ring has an example. $\endgroup$ – Brian M. Scott Oct 28 '13 at 10:28
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Noncommutative domains which satisfy the right Ore condition allow you to build a "right division ring of fractions" in an analogous way to that of the field of fractions for a commutative domain.

This division ring is necessarily not commutative if you pick the domain to be not commutative :) Not commutative right Ore domains are pretty easy to come by: in particular, right Noetherian domains are right Ore. So for example, you could look at the division ring of quotients for $\Bbb H[x]$.

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You can find many examples in:

R. S. Pierce, Associative algebras,Springer-Verlag, 1982

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