\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\\Fall 2005\\Homework 2}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Monday September 19, 2005.}

\centerline{}

\noindent 1. A ring is called {\it Von Neumann regular} if for every $x\in R$ there is a $y\in R$ such that $xyx=x$. Let $R$ be a (commutative) Von Neumann regular ring.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ is an integral domain, then $R$ is a field.
\item[b)] (5 pt) Show that any direct product of fields is Von Neumann regular.
\item[c)] (5 pt) Show that if $\mathfrak{P}$ is a prime ideal in $R$ then $R_{\mathfrak{P}}\cong R/\mathfrak{P}$.
\end{itemize}

\centerline{}

\noindent 2. A ring is called Artinian if it satifies the descending chain on prime ideals (that is given any descending chain of ideals $I_1\supseteq I_2\supseteq I_3\cdots$, there is a natural number $N$ such that $I_n=I_N$ for all $n\geq N$). 
\begin{itemize}
\item[a)] (5 pt) Show that any Artinian domain is a field.
\item[b)] (5 pt) Show that $R$ is Artinian if and only if $R$ is Noetherian and zero-dimensional.
\end{itemize}

\centerline{}

\noindent 3. Let $R$ be an integral domain and consider the ring of formal power series $R[[x]]$.
\begin{itemize}
\item[a)] (5 pt) Show that $U(R[[x]])=\{f(x)\in R[[x]]\vert f(0)\in U(R)\}$.
\item[b)] (5 pt) Show that there is a one-to-one correspondence between the maximal ideals of $R$ and the maximal ideals of $R[[x]]$.
\item[c)] (5 pt) Show that if $R$ is a PID, then $R[[x]]$ is a UFD.
\end{itemize}



\end{document}