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\title{Math 724\\Fall 2005\\Homework 4}
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\centerline{\it Due Friday October 21, 2005.}

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\noindent 1. (5 pt) Show that the domain $V$ is a valuation domain if and only if all of the ideals of $V$ are linearly ordered (can you replace ``ideals" with ``prime ideals" here?).

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\noindent 2. (5pt) Show that if the valuation domain, $V$, is Noetherian  then $\text{dim}(V)\leq 1$.

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\noindent 3. (5 pt) Construct a non-Noetherian, 1-dimensional valuation domain.

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\noindent 4. Let $V$ be a valuation domain and $x$ a nonzero element. 
\begin{itemize}
\item[a)] (5 pt) Show that $x$ is irreducible if and only if $x$ is prime. 
\item[b)] (5 pt) If the valuation domain, $V$, is not a field, then show that $V$ contains a nonzero prime element if and only if the maximal ideal of $V$ is principal.
\item[c)] (5 pt) Show that $V$ is atomic if and only if $V$ is a UFD.
\end{itemize}

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\noindent 5. (5 pt) Given a valuation domain, $V$, construct the complete integral closure of $V$ (hint: there should be two cases).


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