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division_ring [2013/08/05 23:52] nikolaj created |
division_ring [2013/08/05 23:56] nikolaj |
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| @#88DDEE: $\langle X,* \rangle \in \mathrm{group}(X)$ | | | @#88DDEE: $\langle X,* \rangle \in \mathrm{group}(X)$ | | ||
- | | @#55EE55: $\exists (a,b\in X).\ (a\neq b)$ | | + | | @#DDDDDD: $a,b\in X$ | |
+ | |||
+ | | @#55EE55: $\exists a,b.\ (a\neq b)$ | | ||
==== Ramifications ==== | ==== Ramifications ==== | ||
=== Discussion === | === Discussion === | ||
- | A division ring is essentially two groups over a set $X$, one of which is necessarily commutative. The second requirement destinguishes 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. 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]] |