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division_rings [2010/07/29 15:46] (current)
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+=====Division rings=====
+Abbreviation: **DRng**
+====Definition====
+A \emph{division ring} (also called \emph{skew field}) is a [[ring with identity]] $\mathbf{R}=\langle R,+,-,0,\cdot,1 +\rangle$ such that
+
+
+$\mathbf{R}$ is non-trivial:  $0\ne 1$
+
+
+every non-zero element has a multiplicative inverse:  $x\ne 0\Longrightarrow \exists y +(x\cdot y=1)$
+
+Remark:
+The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.
+
+
+==Morphisms==
+Let $\mathbf{R}$ and $\mathbf{S}$ be fields. A morphism from $\mathbf{R}$
+to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism:
+
+$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
+
+Remark:
+It follows that $h(0)=0$ and $h(-x)=-h(x)$.
+
+====Examples====
+Example 1: $\langle\mathcal{Q},+,-,0,\cdot,1\rangle$, the division ring of quaternions with addition, subtraction, zero, multiplication, and one.
+
+
+====Basic results====
+$0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.
+
+====Properties====
+^[[Classtype]]  |first-order |
+^[[Equational theory]]  | |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  |yes |
+^[[Congruence modular]]  |yes |
+^[[Congruence n-permutable]]  |yes, $n=2$ |
+^[[Congruence regular]]  |yes |
+^[[Congruence uniform]]  |yes |
+^[[Congruence extension property]]  | |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]  | |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+====Finite members====
+Every finite division ring is a [[fields]] (i.e. $\cdot$ is commutative).
+J. H. Maclagan-Wedderburn,\emph{A theorem on finite algebras},
+Trans. Amer. Math. Soc.,
+\textbf{6}1905,349--352[[MRreview]]
+
+====Subclasses====
+[[Fields]]
+
+[[Algebraically closed division rings]]
+
+====Superclasses====
+[[Rings with identity]]
+
+
+====References====
+
+[(Ln19xx>
+)]