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division_ring [2013/08/05 23:52] nikolaj created |
division_ring [2013/08/05 23:58] 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: $a,b\in X$ | |
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| + | | @#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. Compatible in the sense of the distributive laws of a ring, which is asymmetrical with respect to "$+$" and "$*$". |
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| + | 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]] | ||
