Bezout domainBezoutDomain2004-04-23 18:50:402005-01-28 00:28:02DefinitionBezout identity% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{Bezout domain} $D$ is an integral domain such that every finitely generated ideal of $D$ is \PMlinkname{principal}{PID}.
\textbf{Remarks}.
\begin{itemize}
\item A PID is obviously a Bezout domain.
\item Furthermore, a Bezout domain is a gcd domain. To see this, suppose $D$ is a Bezout domain with $a,b\in D$. By definition, there is a $d\in D$ such that $(d)=(a,b)$, the ideal generated by $a$ and $b$. So $a\in (d)$ and $b\in (d)$ and therefore, $d\mid a$ and $d\mid b$. Next, suppose $c\in D$ and that $c\mid a$ and $c\mid b$. Then both $a,b\in (c)$ and so $d\in (c)$. This means that $c\mid d$ and we are done.
\item From the discussion above, we see in a Bezout domain $D$, a greatest common divisor exists for every pair of elements. Furthermore, if $\operatorname{gcd}(a,b)$ denotes one such greatest common divisor between $a,b\in D$, then for some $r,s\in D$:
$$\operatorname{gcd}(a,b)=ra+sb.$$
The above equation is known as the \emph{Bezout identity}, or Bezout's Lemma.
\end{itemize}