Division ring
Set
| context | X |
| postulate | ⟨X,+,∗⟩∈divisionRing(X) |
| context | ⟨X,+,∗⟩∈unitalRing(X) |
| context | ⟨X,∗⟩∈group(X) |
| range | a,b∈X |
| postulate | ∃a,b. (a≠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 ⟨X,+,∗⟩ must not be the trivial ring.
Reference
Wikipedia: Division ring, Trivial ring
Parents
Subset of
