division_rings - MathStructures

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=====Division rings=====

Abbreviation: **DRng**
====Definition====
A \emph{division ring} (also called \emph{skew field}) is a [[rings with identity|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>
)]

Trace: • cancellative_commutative_monoids • distributive_residuated_lattices • semiassociative_relation_algebras • commutative_involutive_fl-algebras • generalized_pseudo-effect_algebras • finite_ordered_semigroups_with_zero • distributive_lattices_with_operators • finite_binars_with_identity_and_zero • normal_valued_lattice-ordered_groups • weakly_associative_relation_algebras