gcd domainGcdDomain2004-04-23 18:18:212008-08-23 02:08:22Definitiongcdgreatest common divisorrelatively primelcm domain% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $D$ 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}
Now, a gcd of two elements is in general not unique. However, by definition, any two gcd's of a pair of elements in $D$ are associates of each other. Since the binary relation ``being associates'' of one anther is an equivalence relation (\emph{not} a congruence relation!), we may define \emph{the} gcd of $a$ and $b$ as the set
$$\operatorname{GCD}(a,b):=\lbrace c\in D\mid c\mbox{ is a gcd of }a\mbox{ and }b\rbrace,$$
and, if there is no confusion, denote $\gcd(a,b)$ to be any element of $\operatorname{GCD}(a,b)$.
If $\operatorname{GCD}(a,b)$ contains a unit, then $a$ and $b$ are said to be \emph{relatively prime}. If $a$ is irreducible, then for any $b\in D$, $a,b$ are either relatively prime, or $a\mid b$.
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.
\item Given an integral domain, one can similarly define an lcm of two elements $a,b$: it is an element $c$ such that $a \mid c$ and $b \mid c$, and if $d$ is an element such that $a \mid d$ and $b \mid d$, then $c \mid d$. Then, a
\emph{lcm domain} is an integral domain such that every pair of elements has a lcm. As it turns out, the two notions are equivalent: an integral domain is lcm iff it is gcd.
\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}