A commutative division ring is a field. You can explore field theory if you like, but for now we are going to explore noncommutative division rings.
The classic example of a division ring is the quaternions. A basic understanding of the quaternions will help, since we will refer to them from time to time.
It is logical to start with something simple, like a finite division ring. But there's nothing simple about this topic. Every finite division ring is a field, yet the proof is far from obvious. Read on and see.