\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{color}
\colorlet{darkgreen}{green!70!black}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{midnightblue}{rgb}{.2, .2, .7}
\definecolor{grey}{rgb}{.7, .7, .7}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\definecolor{Green}{rgb}{0,.9,0}
\colorlet{lred}{red!50!white}
\definecolor{lblue}{rgb}{0.60, 0.75, 1}
\colorlet{lgreen}{green!40!white}
\colorlet{lyellow}{yellow!30!white}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}

\colorlet{darkgreen}{green!70!black}
\colorlet{lightred}{red!30!white}
\colorlet{lightblue}{blue!30!white}
\colorlet{lightgreen}{darkgreen!30!white}
\definecolor{lpurple}{rgb}{0.6 0.35 0.8}
\colorlet{darkpurple}{lpurple!70!black}

\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{intersections}

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Ob}{Ob}

\newcommand{\catname}[1]{{\normalfont\mathbf{#1}}}
\newcommand{\Nil}{\catname{Nil}}

\def\<{\langle}
\def\>{\rangle}
\def\into{\hookrightarrow}
\def\longto{\longrightarrow}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\DeclareMathOperator\cl{cl}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\Perm{Perm}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}

\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due November 20 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}

\noindent Unless otherwise specified, assume that $R$ is a commutative
ring with $1$. \medskip


\begin{enumerate}

\item Let $P$ be a non-zero prime ideal.
\begin{enumerate}
\item Show that if $I\cap J\subseteq P$, then $I\subseteq P$ or
  $J\subseteq P$.
\item Show that if $R$ is a PID, then $P$ is maximal. \\
\end{enumerate}



\item For a semigroup $S\subseteq R$ containing no zero divisors, let
  $X=R\times S$ and define a relation where $(a,b)\sim(c,d)$ if
  $ad=bc$.
  \begin{enumerate}
  \item Show that $\sim$ is an equivalence relation on $X$.
  \item Denote the equivalence class of $(a,b)$ by $a/b$ and the set
    of equivalence classes by $R_S$ (called the \emph{localization} of
    $R$ at $S$). Show that $R_S$ is a commutative ring with $1$.
  \item If $a\in S$, show that $\{ra/a\mid r\in R\}$ is a subring of
    $R_S$ and that $r\mapsto ra/a$ is a monomorphism, so that $R$ can
    be identified with a subring with $R_S$.
  \item Show that every $s\in S$ is a unit in $R_S$.
  \item Give a ``universal'' definition for $R_S$ and show that it is
    unique up to isomorphism. \\
\end{enumerate}



\item A \emph{local ring} is a commutative ring with identity which
  has a unique maximal ideal. 
  \begin{enumerate}
  \item Show that if $x$ is contained in every maximal ideal of $R$,
    then $1+x$ is a unit.
  \item Prove that a commutative ring is local if and only if its set of
    non-units is an ideal.
  \item If $P$ is a prime ideal of an integral domain $R$, show that
    $S=R\setminus P$ is a multiplicative semigroup and $R_S$ is local. \\
  \end{enumerate}



\item For each of the rings $R$ and multiplicative semigroups $S$, construct the localization $R_S$ and describe the elements, units, and all maximal ideals. 
  \begin{enumerate}
  \item $R=\Z$ and $S=R\setminus(5)$. 
  \item $R=\Z$ and $S=(5)\setminus\{0\}$.
  \item $R=\Z$ and $S=\{5^k\mid k\in\N\}$.
  \item $R=\Z[x]$ and $P=R\setminus (x)$. \\
    \end{enumerate}



\item Find the field of fractions of the following rings. Prove your claims.
\begin{enumerate}
\item $R$ is any field.
\item $R=2\Z$.
\item $R=\Z[i]:=\{a+bi\mid a,b\in\Z\}$. \\
\end{enumerate}



\item Use Zorn's lemma to show that the ring $\R$ contains a subring
  $A$ containing $1$ that is maximal with respect to the property that
  $1/2\not\in A$, and then find its field of fractions.


\end{enumerate}


\end{document}




