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\title{Math 724\\Fall 2010\\Homework 5}
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\centerline{\it Due Monday, April 23, 2010}

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\noindent 1. (5 pt) We have (or will have) shown that if $R[x]$ is an HFD, then $R$ must be integrally closed. Can we replace ``integrally closed" with ``completely integrally closed"?

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\noindent 2. We say that a domain, $R$, is an AP-domain if every atom is prime.  
\begin{itemize}
\item[a)] (5 pt) Show that any GCD-domain is an AP-domain.
\item[b)] (5 pt) Show that in the class of atomic domains, the notions of AP-domain, GCD-domain, and UFD are all equivalent.
\end{itemize}

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\noindent 3. We will (unfortunately) say that $R$ is a U-UFD if every nonzero nonunit that can be factored into irreducibles does so uniquely.
\begin{itemize}
\item[a)] (5 pt) Show that any AP-domain is a U-UFD.
\item[b)] (5 pt) Show that if $R$ is a domain with precisely one irreducible and this irreducible is not prime, then $R$ is a U-UFD that is not an AP-domain.
\end{itemize}

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\noindent 4. We say that $R$ is a CK-domain if $R$ is atomic and has only finitely many irreducibles (up to associates). We use the notation CK-$n$ to refer to a CK-domain with precisely $n$ irreducibles.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ is CK-$n$ for $n\leq 2$ then $R$ is a PID (and hence a UFD).
\item[b)] (5 pt) Find an example of a CK-3 domain that is not a UFD.
\end{itemize}

  
 





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