Mathematical Structures: Division rings

# Division rings

Difference (from prior major revision) (author diff)

Changed: 32c32
 A division ring (also called skew field) is a \href{Rings_with_identity.pdf}{rings with identity} $\mathbf{R}=\langle R,+,-,0,\cdot,1  A division ring (also called skew field) is a \href{Rings_with_identity.pdf}{ring with identity}$\mathbf{R}=\langle R,+,-,0,\cdot,1

Changed: 89,91c89
 \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr}  \begin{finite_members} Changed: 95,96c93  61905,349--352\href{"http://www.ams.org/mathscinet-getitem?mr=1 500 717:"}{MRreview} \end{array}$
 61905,349--352\href{http://www.ams.org/mathscinet-getitem?mr=1 500 717}{MRreview}

http://mathcs.chapman.edu/structuresold/files/Division_rings.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Division rings}

\abbreviation{DRng}
\begin{definition}
A \emph{division ring} (also called \emph{skew field}) is a \href{Rings_with_identity.pdf}{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:  $x\ne 0\implies \exists y (x\cdot y=1)$

Remark:
The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.

\end{definition}

\begin{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 $h(-x)=-h(x)$.

\end{morphisms}
\begin{basic_results}
$0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.
\end{basic_results}
\begin{examples}
\begin{example}
$\langle\mathcal{Q},+,-,0,\cdot,1\rangle$, the division ring of quaternions with addition, subtraction, zero, multiplication, and one.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & first-order\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members}
Every finite division ring is a \href{Fields.pdf}{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\href{http://www.ams.org/mathscinet-getitem?mr=1 500 717}{MRreview}
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Fields.pdf}{Fields}

\href{Algebraically_closed_division_rings.pdf}{Algebraically closed division rings}

\end{subclasses}
\begin{superclasses}\

\href{Rings_with_identity.pdf}{Rings with identity}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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