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\title{Math 721\\Spring 2011\\Final Exam}
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\begin{center}
{\it Due before I have to chase you to Pakistan. As is usual on exams, I am the only biological resource that you should use.}
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\noindent 1. Let $R$ be a domain with quotient field $K$ and $\overline{R}$ the integral closure of $R$. $V$ will denote a valuation overring of $R$.
\begin{itemize}
\item[a)] Show that $\overline{R}$ is integrally closed.
\item[b)] Show that $\overline{R}\subseteq\bigcap_{R\subseteq V\subseteq K} V$.
\item[c)] Now show that $\overline{R}=\bigcap_{R\subseteq V\subseteq K} V$. 
\end{itemize}

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\noindent 2. Let $R$ be a domain with quotient field $K$. An element $x\in K$ is said to be {\it almost integral} if there is a nonzero $r\in R$ such that $rx^n\in R$ for all $n\in\mathbb{N}$. We say that a domain is {\it completely integrally closed} if it contains all of its almost integral elements.
\begin{itemize}
\item[a)] Show that if $x\in K$ is integral over $R$, then $x$ is almost integral over $R$.
\item[b)] Give an example of an element that is almost integral, but not integral.
\item[c)] Show that if $R$ is Noetherian, then any almost integral element over $R$ is integral over $R$.
\item[d)] Let $V$ be a valuation domain that is not a field. Show that $V$ is completely integrally closed if and only if $V$ is one-dimensional (that is, every nonzero prime ideal is maximal).
\end{itemize}

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\noindent 3. We say that $R$ is a Pr\"{u}fer domain if $R_{\mathfrak{P}}$ is a valuation domain for all prime ideals $\mathfrak{P}\subseteq R$. We say that $R$ is a Bezout domain if every finitely generated ideal is principal.
\begin{itemize}
\item[a)] Show that $V$ is a valuation domain if and only if $V$ is a quasilocal Bezout domain.
\item[b)] Show that $R$ is a Pr\"{u}fer domain if and only if every finitely generated ideal of $R$ is invertible.
\end{itemize}

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