History of Commutative rings

 
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 \begin{document}
 \textbf{\Large Commutative rings}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Commutative_rings}{edit}
 
 \abbreviation{CRng}
 \begin{definition}
 A \emph{commutative ring} is a \href{Rings.pdf}{rings} $\mathbf{R}=\langle R,+,-,0,\cdot\rangle$ such that
 
 $\cdot$ is commutative:  $x\cdot y=y \cdot x$
 
 
 Remark:  $Idl(R)=\{ all ideals of R\}$
 
 $I$ is an ideal if $a,b\in I\implies a+b\in I$
  
 and $\forall r \in R\ (r\cdot I\subseteq I)$
 
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{R}$ and $\mathbf{S}$ be commutative rings with identity. A morphism from $\mathbf{R}$
 to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism: 
 
 $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$
 
 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}
 $\langle\mathbb{Z},+,-,0,\cdot\rangle$, the ring of integers with addition, subtraction, zero, and multiplication.
 \end{example}
 \end{examples}
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & variety\\\hline
 Equational theory & decidable\\\hline
 Quasiequational theory & \\\hline
 First-order theory & undecidable\\\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 $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &2\\
 f(3)= &2\\
 f(4)= &9\\
 f(5)= &2\\
 f(6)= &4\\
 [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A037289 Finite commutative rings in the Encyclopedia of Integer Sequences]
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Commutative_rings_with_identity.pdf}{Commutative rings with identity} 
 
 \href{Fields.pdf}{Fields} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Rings.pdf}{Rings} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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