\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[all]{xy}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6in
\begin{document}
\author{}
\title{Math 724\\Summer 2010\\Homework 4}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Friday, August 6, 2010.}

\centerline{}

\noindent 1. (5 pt) For this first one, you may use the fact that is $I$ is a fractional ideal, then there are elements $x,y\in I$ such that $R:(R:I)=R:(R:(Rx+Ry))$. The problem is to explain why if $I$ is divisorial then there are elements $u,v\in (R:I)\setminus \{0\}$ such that $I=Ru^{-1}\bigcap Rv^{-1}$. 


\centerline{}

\noindent 2. Let $R$ be an integral domain with quotient field $K$. We say that the element $\omega\in K$ is $\Omega$-almost integral if $r\omega\in R$ implies that we can find a positive integer $b$ such that $r^b\omega^n\in R$ for all $n\geq 0$. Show that following.
\begin{itemize}
\item[a)] (5 pt) If $\omega$ is $\Omega$-almost integral, then $\omega$ is almost integral. 
\item[b)] (5 pt) Give an example of an almost integral element that is not $\Omega$-almost integral.
\item[c)] (5 pt) Show that $V$ is a valuation domain, then $V$ is $\Omega$-almost integrally closed.
\item[d)] (5 pt) Show that $D$ is a Pr\"{u}fer domain, then $D$ is $\Omega$-almost integrally closed.
\end{itemize}

\end{document}