===== Division ring =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $X$ |
| @#55EE55: postulate   | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ |
| @#55CCEE: context     | @#55CCEE: $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ |
| @#55CCEE: context     | @#55CCEE: $\langle X,* \rangle \in \mathrm{group}(X)$ |
| @#DDDDDD: range       | @#DDDDDD: $a,b\in X$ |
| @#55EE55: postulate   | @#55EE55: $\exists a,b.\ (a\neq b)$ |

==== Ramifications ====
=== Discussion ===
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.
==== Reference ====
Wikipedia: [[http://en.wikipedia.org/wiki/Division_ring|Division ring]], [[http://en.wikipedia.org/wiki/Trivial_ring|Trivial ring]]
==== Parents ====
=== Subset of ===
[[Unital ring]]