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| Title of object: |
gcd domain |
| Canonical Name: |
GcdDomain |
| Type: |
Definition |
| Created on: |
2004-04-23 18:18:21 |
| Modified on: |
2007-05-11 01:59:53 |
| Classification: |
msc:13G05 |
Revision comment (for changes between this and next version):
| Changes for correction #11941 ('please add to defines list'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Let $D$ be a commutative ring with $1\neq 0$. A gcd (greatest common divisor) of two elements $a, b \in D$, is an element $d \in D$ such that:
\begin{enumerate}
\item
$d\mid a$ and $d\mid b$,
\item
if $c\in D$ with $c\mid a$ and $c\mid b$, then $c\mid d$.
\end{enumerate}
Any two gcd's of a pair of elements in $D$ are associates of each other. Therefore we can speak of \emph{the} gcd of $a$ and $b$ with the knowledge that any two such gcd's are the same by a product of a unit. We denote the gcd of elements $a, b \in D$ to be $\gcd(a,b)$.
An integral domain $D$ is called a \emph{gcd domain} if any two elements of $D$, not both zero, have a gcd.
\textbf{Remarks}
\begin{itemize}
\item A unique factorization domain, or UFD is a gcd domain, but the converse is not true.
\item A Bezout domain is always a gcd domain. A gcd domain $D$ is a Bezout domain if $\gcd(a,b) = ra+sb$ for any $a, b \in D$ and some $r, s \in D$.
\item In a gcd domain, an irreducible element is a prime element.
\item A gcd domain is integrally closed. In fact, it is a Schreier domain.
\end{itemize}
The following diagram indicates how the different domains are related:
\begin{center}
\begin{tabular}{c c c c c}
\PMlinkname{Euclidean domain}{EuclideanRing} & $\Longrightarrow$ & PID & $\Longrightarrow$ & UFD \\
& & & & \\
& & $\Downarrow$ & & $\Downarrow$ \\
& & & & \\
& & Bezout domain & $\Longrightarrow$ & gcd domain \\
\end{tabular}
\end{center} |
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