| context | $X$ |
| postulate | $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ |
| context | $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ |
| context | $\langle X,* \rangle \in \mathrm{group}(X)$ |
| range | $a,b\in X$ |
| postulate | $\exists a,b.\ (a\neq b)$ |
A division ring is essentially two compatible groups over a set $X$, one of which is necessarily commutative. Compatible in the sense of the distributive laws of a ring, which is asymmetrical with respect to “$+$” and “$*$”.
The second requirement distinguishes the division ring from a unital ring by inverses with respect to the multiplication $*$. The last statement says that $\langle X,+,* \rangle$ must not be the trivial ring.
Wikipedia: Division ring, Trivial ring