Mathematical Structures: Principal Ideal Domains

# Principal Ideal Domains

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 %%run pdflatex % \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 Principal Ideal Domain} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Principal_Ideal_Domain}{edit} \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 The American Mathematical Monthly 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} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \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} %
 %%run pdflatex % \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 Principal Ideal Domain} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Principal_Ideal_Domain}{edit} \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 The American Mathematical Monthly 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} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \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} %

http://mathcs.chapman.edu/structuresold/files/Principal_Ideal_Domains.pdf
%%run pdflatex

%


\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 Principal Ideal Domain}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Principal_Ideal_Domain}{edit}

\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}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\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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Last edited July 10, 2004 10:49 am by Jipsen (diff)
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