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\title{Math 724\\Summer 2010\\Homework 1}
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\centerline{\it Due Monday, June 28, 2010.}

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\noindent 1. Let $V$ be a valuation domain. Prove the following.
\begin{itemize}
\item[a)] (5 pts) Show that $V$ is integrally closed.
\item[b)] (5 pts) Show that every overring of $V$ is a valuation domain.
\item[c)] (5 pts) Show that every overring of $V$ is a localization of $V$.
\end{itemize}

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\noindent 2. (5 pts) Let $\mathbb{F}$ be a field. Find an overring of $\mathbb{F}[x,y]$ that is not integrally closed or prove that no such overring can exist.

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\noindent 3. Let $R$ be an integral domain, $I\subseteq R$ a nonzero ideal, and $S$ a multiplicatively closed subset of $R$.
\begin{itemize}
\item[a)] (5 pts) Show that if $I$ is an invertible ideal of $R$ then $I_S$ is an invertible ideal of $R_S$.
\item[b)] (5 pts) Show that if $I$ is finitely generated then $I$ is an invertible ideal of $R$ if and only if $I_{\mathfrak{M}}$ is principal for all $\mathfrak{M}\in\text{MaxSpec}(R)$.
\end{itemize}  






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