\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 13 $|$ Due December 7 (Friday)
}} \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 For an ideal $I$ of $R[x]$ and $m\in\N$, let $I(m)$ denote the
  set of leading coefficients of degree-$m$ polynomomials in $I$,
  together with $0$.
  \begin{enumerate}
  \item Show that $I(m)$ is an ideal in $R$.
  \item Show that $I(m)\subseteq I(m+1)$ for all $m$.
  \item If $J$ is an ideal with $I\subseteq J$, show that
    $I(m)\subseteq J(m)$ for all $m$. \medskip
  \end{enumerate}



\item Consider the following rings $R_i$, for $i=1,\dots,6$, which are
  additionally $\C$-vector spaces:

  \hspace{15mm} $R_1 = \C[x]/(x^3-1)$
    
  \hspace{15mm} $R_2 = \C \times \C \times \C$
    
  \hspace{15mm} $R_3 = $ the ring of upper triangular $2 \times 2$ matrices over
    $\C$
    
  \hspace{15mm} $R_4 = \C[x]/(x-1) \times \C[x]/(x+i)\times\C[x]/(x-i)$
    
  \hspace{15mm} $R_5 = \C[x]/(x^2+1) \times \C[x]/(x-1)$
    
  \hspace{15mm} $R_6 = \C[x]/(x+1)^2 \times \C[x]/(x-1)$.

    \begin{enumerate} 
    \item Compute the dimension of each $R_i$ as a $\C$-vector space by
      giving an explicit basis.
    \item Partition the rings $R_1,\dots,R_6$ into isomorphism classes. \medskip
    \end{enumerate}



\item 
  \begin{enumerate}
  \item Solve the congruences
    \[
    x\equiv 1 \pmod{8}\,,\qquad
    x\equiv 3 \pmod{7}\,,\qquad
    x\equiv 9 \pmod{11}\,
    \]
    simultaneously for $x$ in the ring $\Z$ of integers.
  \item Solve the congruences
    \[
    x\equiv i \pmod{i+1}\,,\qquad
    x\equiv 1 \pmod{2-i}\,,\qquad
    x\equiv 1+i \pmod{3+4i}\,
    \]
    simultaneously for $x$ in the ring $R_{-1}$ of Gaussian integers. 
  \item Solve the congruences
    \[
    f(x)\equiv 1 \pmod{x-1}\,,\qquad
    f(x)\equiv x \pmod{x^2+1}\,,\qquad
    f(x)\equiv x^3 \pmod{x+1}\,
    \]
    simultaneously for $f(x)$ in $F[x]$, where $F$ is a field in which
    $1+1\neq 0$. \medskip
  \end{enumerate}



\item An element $e\in R$ is called an \emph{idempotent} if $e^2=e$, and two
  nonzero idempotents $e_1,e_2$ are called an \emph{orthogonal pair} if
  $e_1+e_2=1$ and $e_1e_2=0$.
  \begin{enumerate}
  \item Show that the following are equivalent:
    \begin{enumerate}[(i)]
    \item $R$ contains an idempotent different from 0 and 1.
    \item $R$ contains an orthogonal pair of idempotents.
    \item $R\cong R_1\times R_2$ for some rings $R_1$ and $R_2$.
    \end{enumerate}
  \item Give an example of a non-orthogonal pair of distinct idemponents.
  \item Find all idempotents in the ring $\Z/20\Z$.
  \end{enumerate}


\end{enumerate}


\end{document}






