Unique factorization domains

http://mathcs.chapman.edu/structuresold/files/Unique_factorization_domains.pdf

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\begin{document}
\textbf{\Large Unique Factorization Domains}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Unique_Factorization_Domains}{edit}

\abbreviation{UFDom}
\begin{definition}
A \emph{unique factorization domain} is an \href{Integral_domains.pdf}{integral domains} $D$ such that


every element is a product of irreducibles:  $\forall a\in D \exists p_1,...,p_r\in D, n_1,...,n_r\in \mathbb{N}$ such that $a=p_1^{n_1}\cdotp_2^{n_2}...p_r^{n_r}$ and 
$p_i$ is irreducible for $i=1,\ldots,r$


the product is unique up to associates:  $\forall \mbox{ irreducibles } p_i,q_j$ if 
$a=p_1^{n_1}\cdot p_2^{n_2}...p_r^{n_r}=q_1^{m_1}\cdot q_2^{m_2}...q_s^{m_s}$
then $r=s$ and each $p_i$ is an associate of some $q_j$
\end{definition}

\begin{morphisms}

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\mathbb{Z}[x]$, the ring of polynomials with integer coefficients.
\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/}
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\begin{subclasses}\ 

\href{Principal_Ideal_Domains.pdf}{Principal Ideal Domains} 

\end{subclasses}
\begin{superclasses}\ 

\href{Integral_domains.pdf}{Integral domains} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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