Examples of a non-commutative division ring

abstract-algebra ring-theory quaternions

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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