|
%%run pdflatex %
|
|
%%run pdflatex %
|
![[Home]](/structures.gif)
\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\addtolength{\oddsidemargin}{-.5in}
\addtolength{\evensidemargin}{-.5in}
\addtolength{\textwidth}{1in}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}
\begin{document}
\textbf{\Large Regular rings}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Regular_rings}{edit}
\abbreviation{RRng}
\begin{definition}
A \emph{regular ring} is a \href{Rings_with_identity.pdf}{rings with identity} $\mathbf{R}=\langle R,+,-,0,\cdot,1
\rangle $ such that
every element has a pseudo-inverse: $\forall x\exists y(x\cdot y\cdot x=x)$
\end{definition}
\begin{morphisms}
Let $\mathbf{R}$ and $\mathbf{S}$ be regular rings. 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{examples}
\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 & no\\\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} $f(n)=$ number of members of size $n$.
$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\
\href{Division_rings.pdf}{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}
%