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\title{Math 725\\Fall 2006\\Homework 6}
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\centerline{\it Due Monday, December 4, 2006.}

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\noindent 1. (5 pt) Show that if $R[x]$ is an AP-domain, then $R$ must be integrally closed (is the converse true?).

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\noindent 2. Let $V$ be a valuation domain.
\begin{itemize}
\item[a)] (5 pt) Show that $V$ is an AP-domain.
\item[b)] (5 pt) Show that $V$ has atoms if and only if the maximal ideal of $V$ is principal.
\item[c)] (5 pt) Give an example of an atomic valuation domain.
\item[d)] (5 pt) Give an example of a nonatomic valuation domain with atoms.
\item[e)] (5 pt) Give an example of an antimatter valuation domain of dimension 1 and an antimatter valuation domain of dimension greater than 1.
\end{itemize}


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\noindent 3. (5 pt) Give an example of a non-integrally closed AP domain and use this to give an example of an AP-domain, $R$, such that $R[x]$ is not an AP-domain.
 




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