Does there exist a noncommutative division ring $D$ (i. e. a field except that commutativity of multiplication is violated, e. g. the quaternions) which is also an ordered ring?
Since most examples of division rings I am aware of are constructed in a manner very similarto the quaternions and thus not orderable, I would be very curious to see an example of such a ring.
As I understand it a skew Laurent series ring (see here in the examples section) is such a ring. According to page 10 of "A First Course in Noncommutative Rings" by Tsi-Yuen Lam, Hilbert was the first to note this.