History of Regular rings

 
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 \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 $n$.
 
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 \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}
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