Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
division_ring [2013/08/05 23:56]
nikolaj
division_ring [2017/03/25 15:59]
nikolaj
Line 1: Line 1:
 ===== Division ring ===== ===== Division ring =====
-==== Definition ​==== +==== Set ==== 
-| @#88DDEE: $X$ | +| @#55CCEE: context ​    | @#55CCEE: $X$ | 
- +| @#55EE55: postulate ​  | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ | 
-| @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ | +| @#55CCEE: context ​    | @#55CCEE: $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ | 
- +| @#55CCEE: context ​    | @#55CCEE: $\langle X,* \rangle \in \mathrm{group}(X)$ | 
-| @#88DDEE: $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ | +| @#DDDDDD: range       | @#DDDDDD: $a,b\in X$ | 
-| @#88DDEE: $\langle X,* \rangle \in \mathrm{group}(X)$ | +| @#55EE55: postulate ​  | @#55EE55: $\exists a,b.\ (a\neq b)$ |
- +
-| @#DDDDDD: $a,b\in X$ | +
- +
-| @#55EE55: $\exists a,b.\ (a\neq b)$ |+
  
 ==== Ramifications ==== ==== Ramifications ====
 === Discussion === === Discussion ===
-A division ring is essentially two compatible groups over a set $X$, one of which is necessarily commutative. 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.+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 ==== ==== Reference ====
 Wikipedia: [[http://​en.wikipedia.org/​wiki/​Division_ring|Division ring]], [[http://​en.wikipedia.org/​wiki/​Trivial_ring|Trivial ring]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Division_ring|Division ring]], [[http://​en.wikipedia.org/​wiki/​Trivial_ring|Trivial ring]]
-==== Context ​====+==== Parents ​====
 === Subset of === === Subset of ===
 [[Unital ring]] [[Unital ring]]
Link to graph
Log In
Improvements of the human condition