# Differences

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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> | ||

+ | )] |

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