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\title{Math 725\\Fall 2005\\Homework 1}
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\centerline{\it Due Friday, September 8, 2006.}

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\noindent 1. Let $R$ be a domain, and $a,b\in R$. We define the {\it greatest common divisor} of $a$ and $b$ to be a common divisor $d:=\text{gcd}(a,b)$ with the property that if $x$ is any other common divisor of $a$ and $b$ then $x$ divides $d$. We also define the {\it least common multiple} to be a common multiple $L:=\text{lcm}(a,b)$ with the property that if $y$ is any other common multiple of $a$ and $b$ then $L$ divides $y$. 

\begin{itemize}
\item[a)] (5 pt) Give an example of a domain, $R$, and two elements $a,b\in R$ such that $\text{gcd}(a,b)$ exists but $\text{lcm}(a,b)$ does not.
\item[b)] (5 pt) Show that if $\text{lcm}(a,b)$ exists, then so does $\text{gcd}(a,b)$.
\end{itemize}

\centerline{}

\noindent 2. A {\it GCD domain} is an integral domain in which every two (nonzero) elements have a greatest common divisor. Show that in a GCD domain the following hold.

\begin{itemize}
\item[a)] (5 pt) $x(\text{gcd}(a,b))=\text{gcd}(xa,xb)$.
\item[b)] (5 pt) If $\text{gcd}(a,b)=d$ then $\text{gcd}(\frac{a}{d},\frac{b}{d})=1$.
\item[c)] (5 pt) If $\text{gcd}(x,a)=1$ and $\text{gcd}(x,b)=1$ then $\text{gcd}(x,ab)=1$.
\item[d)] (5 pt) If $\text{gcd}(x,a)=1$ and $x$ divides $ab$ then $x$ divides $b$.
\item[e)] (5 pt) Which of the above hold in a general integral domain?
\end{itemize}



\centerline{}

\noindent 3. A Bezout domain is a domain where every finitely generated ideal is principal.
\begin{itemize}
\item[a)] (5 pt) Show $R$ is a valuation domain if and only if $R$ is a quasi-local Bezout domain.
\item[b)] (5 pt) Show that any Bezout domain is a GCD domain.
\item[c)] (5 pt) Show that any UFD is a GCD domain.
\item[d)] (5 pt) Give examples to show that the classes of Bezout domains and UFDs are distinct.
\end{itemize}




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