I think division is normally framed rather as the existence of multiplicative inverses for all non-zero elements of the ring. That is, R is a division ring if

I think division is normally framed rather as the existence of multiplicative inverses for all non-zero elements of the ring. That is, R is a division ring if a) there exists a 1 in R b) for all x in R, there exists y in R such that x * y = 1. I'm pretty sure these are equivalent. Proof: (==>) Let a = 1, and b in R be arbitrary. Then there exists a c such that b * c = 1. So inverses exist! (<==) Let a and b in R be arbitrary. Then a = a * b^-1 * b, so c = a * b^-1. QED.