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division_ring [2013/08/05 23:56] nikolaj |
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| ===== Division ring ===== | ===== Division ring ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
| - | | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ | | + | | @#55EE55: postulate | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ | |
| - | | @#88DDEE: $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ | | + | | @#55CCEE: context | @#55CCEE: $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ | |
| - | | @#88DDEE: $\langle X,* \rangle \in \mathrm{group}(X)$ | | + | | @#55CCEE: context | @#55CCEE: $\langle X,* \rangle \in \mathrm{group}(X)$ | |
| - | | @#DDDDDD: $a,b\in X$ | | + | | @#DDDDDD: range | @#DDDDDD: $a,b\in X$ | |
| - | | @#55EE55: $\exists a,b.\ (a\neq b)$ | | + | | @#55EE55: postulate | @#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]] | ||
