Mathematical Structures: Principal Ideal Domains

# Principal Ideal Domains

http://mathcs.chapman.edu/structuresold/files/Principal_Ideal_Domains.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Principal Ideal Domain}

\abbreviation{PIDom}
\begin{definition}
A \emph{principal ideal domain} is an \href{Integral_domains.pdf}{integral domains} $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ in which

every ideal is principal:  $\forall I \in Idl(R)\ \exists a \in R\ (I=aR)$

Ideals are defined for \href{Commutative_rings.pdf}{commutative rings}
\end{definition}

\begin{morphisms}
\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
${a+b\theta | a,b\in Z, \theta=\langle 1+ \langle-19\rangle^{1/2}\rangle/2}$ is a Principal Ideal Domain that is not an \href{Euclidean_domains.pdf}{Euclidean domains}

See Oscar Campoli's "A Principal Ideal Domain That Is Not a Euclidean Domain" in <i>The American Mathematical Monthly</i> 95 (1988): 868-871
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & Second-order\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & \\\hline
Residual size & \\\hline
Congruence distributive & \\\hline
Congruence modular & \\\hline
Congruence n-permutable & \\\hline
Congruence regular & \\\hline
Congruence uniform & \\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &1\\ f(5)= &1\\ f(6)= &0\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

\href{Euclidean_domains.pdf}{Euclidean domains}

\end{subclasses}
\begin{superclasses}\

\href{Unique_factorization_domains.pdf}{Unique factorization domains}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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