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Division ring


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.



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