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 no unbounded yes yes yes, $n=2$ yes yes

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}1905,349–352MRreview