\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usepackage{tkz-graph}
\usepackage{tikz-cd}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{bm}

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


% Useful math macros
\def\ds{\displaystyle}
\def\bold{\bm}     % Change ``bm'' to ''mathbf'' for upright boldface.

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}
\newcommand{\vvvvv}[5]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{bmatrix}}
\def\<{\langle}
\def\>{\rangle}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}
\def\Dot{\bm{.}}

%\def\subgroup{\leqslant}
%\def\nsubgroup{\nleqslant}
%\def\normal{\leqslant}
%\def\nnormal{\nleqslant}

\newcommand{\rddots}{\rotatebox{90}{$\ddots$}}
\def\setmin{-}

\def\Sum{\sum\limits}
\def\Prod{\prod\limits}
\def\Cap{\mathop\cap\limits}
\def\Cup{\mathop\cup\limits}
\def\Times{\mathop\times\limits}
\def\Oplus{\mathop\oplus\limits}
\def\Lim{\mathop\lim\limits}

\def\And{\wedge}
\def\Or{\vee}
\def\Not{\neg}

\def\longto{\longrightarrow}
\def\into{\hookrightarrow}
\def\longinto{\longhookrightarrow}

\def\A{\mathbb{A}}
\def\C{\mathbb{C}}
\def\F{\mathbb{F}}
\def\HH{\mathbb{H}}
\def\N{\mathbb{N}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}

\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Lcm}{lcm}
\DeclareMathOperator{\del}{\nabla}
\DeclareMathOperator{\Trace}{tr}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator\Perm{Perm}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\orb{orb}
\DeclareMathOperator\Orb{Orb}
\DeclareMathOperator\stab{stab}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\fix{fix}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\Syl{Syl}
\DeclareMathOperator\Pre{Pre}
\DeclareMathOperator\Ker{Ker}
\DeclareMathOperator\Sym{Sym}
\DeclareMathOperator\Mat{Mat}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator\AGL{AGL}
\DeclareMathOperator\PGL{PGL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\PSU{PSU}
\DeclareMathOperator\PSP{PsP}
\DeclareMathOperator\Heis{Heis}
\DeclareMathOperator\Dih{D}
\DeclareMathOperator\Dic{Dic}
\DeclareMathOperator\SA{SA}
\DeclareMathOperator\SD{SD}
\DeclareMathOperator\Mod{Mod}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Deg{Deg}

\DeclareMathOperator\Rect{\mathbf{Rect}}
\DeclareMathOperator\Tri{\mathbf{Tri}}
\DeclareMathOperator\Coin{\mathbf{Coin}}
\DeclareMathOperator\BOX{\mathbf{Box}}
\DeclareMathOperator\Light{\mathbf{Light}}
\DeclareMathOperator\Sq{\mathbf{Sq}}

% Colors 
\usepackage{xcolor}
\usepackage{color}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{grey}{rgb}{.7, .7, .7}
\newcommand{\balert}[1]{\textcolor{blue}{#1}}
\newcommand{\galert}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\palert}[1]{\textcolor{amethyst}{#1}}
\newcommand{\Alert}[1]{\textcolor{Red}{#1}}
\newcommand{\Balert}[1]{\textcolor{Blue}{#1}}
\newcommand{\Galert}[1]{\textcolor{darkgreen}{#1}}

%\hypersetup{colorlinks=true,linkcolor=darkblue,anchorcolor=darkblue,
%  citecolor=darkblue,pagecolor=darkblue,urlcolor=darkblue,
%  pdftitle={},pdfauthor={}
%}                               %% THIS CAUSES COMPILER ERROR
\colorlet{lightred}{red!30!white}
\colorlet{lightblue}{blue!30!white}
\colorlet{lightyellow}{yellow!30!white}
\colorlet{keylime}{green!10!white}
\colorlet{darkgreen}{green!50!black}
\colorlet{midgreen}{green!70!black}
\colorlet{lightgreen}{darkgreen!30!white}
\colorlet{lightpurple}{violet!60!white}
\colorlet{darkgrey}{black!70}
\colorlet{midgrey}{black!50}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\definecolor{Green}{rgb}{0,.9,0}
\definecolor{Gold}{rgb}{.83,.67,.08}

\definecolor{conj-purple}{HTML}{E4B7FF}
\definecolor{conj-orange}{HTML}{FFD4B7}


\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}{0.6 0.35 0.8}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{faded}{rgb}{.75,.75,.75}
\definecolor{midgrey}{rgb}{.5,.5,.5}
%\definecolor{midgrey}{rgb}{.35, .35, .35}
%\definecolor{amethyst}{rgb}{0.6 0.4 0.8}
\definecolor{amethyst}{rgb}{0.54 0.17 0.89}
\definecolor{amber}{rgb}{1.0 0.49 0.0}
\definecolor{turq}{rgb}{0.12 0.69 0.71}


\colorlet{lightestblue}{blue!15!white}
\colorlet{midblue}{blue!45!white}

\colorlet{lightestred}{red!15!white}
\colorlet{midred}{red!45!white}


% COLORS FOR MULTIPLICATION TABLES  
\colorlet{tred}{red!70!white}
\colorlet{tblue}{blue!60!white}     %(higher number is more faded)
\colorlet{tgreen}{green!75!black}   %(higher number is brighter)
\colorlet{tyellow}{yellow!60!white}
\colorlet{tlime}{green!80!white}
\colorlet{tpink}{pink}
\colorlet{tpurple}{lpurple}
\colorlet{torange}{orange}


\newcommand*{\n}{5}   % counter for Cayley tables

%% CAYLEY DIAGRAM GLOBAL DEFAULTS
\tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm]
\tikzstyle{r} = [draw, very thick, Red, -stealth]        % Red --->
\tikzstyle{b} = [draw, very thick, Blue, -stealth]       % Blue --->
\tikzstyle{g} = [draw, very thick, darkgreen, -stealth]  % Green --->
\tikzstyle{p} = [draw, very thick, amethyst, -stealth]     % Purple --->
\tikzstyle{rr} = [draw, very thick, Red]                 % Red ----
\tikzstyle{bb} = [draw, very thick, Blue]                % Blue ----
\tikzstyle{gg} = [draw, very thick, darkgreen]           % Green ----
\tikzstyle{oo} = [draw, very thick, orange]           % Orange ----
\tikzstyle{pp} = [draw, very thick, amethyst]           % Purple ----
\tikzstyle{tt} = [draw, very thick, turq]           % Turquoise ----
\tikzstyle{yy} = [draw, very thick, Gold]           % Gold ----


\tikzstyle{R} = [draw, very thick, Red, -stealth']        % Red --->
\tikzstyle{B} = [draw, very thick, Blue, -stealth']       % Blue --->
\tikzstyle{G} = [draw, very thick, darkgreen, -stealth']  % Green --->

\tikzset{r/.style={draw, very thick, Red, -stealth}}        % Red --->
\tikzset{R/.style={draw, very thick, Red, -stealth'}}        % Red --->

\tikzset{v/.style={circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm}}

\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 April 25 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Spring 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

  %%--------------------------------------------------------------------

\item For each of the following rings $R$, determine the zero divisors
  and the set $U(R)$ of units.
  \begin{enumerate}
  \item The set $\mathcal{C}^1$ of continuous real-valued functions
    $f\colon\R\to\R$.
  \item The polynomial ring $\R[x]$.
    %\item The set of upper-triangular $n\times n$ matrices over $\R$. 
  \item $\Z\times\Z$, where addition and multiplication are defined
    componentwise.
  \item $\R\times\R$, where addition and multiplication are defined
    componentwise. \bigskip
  \end{enumerate}
  
  %%------------------------------------------------------------------
  
\item There is a unique finite field $\F_q$ of order $q=p^k$ for
  every prime $p$ and $k\in\N$. For all other $q\in\N$,
  there is no finite field of order $q$. For each of the fields
  $\F_4$, $\F_5$, and $\F_8$, the Cayley diagrams for addition and
  multiplication are shown below, overlayed on the same set of
  nodes. The solid arrows are the Cayley diagrams for addition and the
  dashed arrows are the Cayley diagrams for multiplication. \vspace{-6mm}
  
  \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=5mm] 
  \tikzstyle{g-v} = [circle, draw, fill=darkgreen,inner sep=0pt,
    minimum size=6mm] 
  %\tikzstyle{b} = [draw,very thick, Blue]
  %\tikzstyle{g} = [draw,very thick, darkgreen]
  \tikzstyle{g-dashed} = [draw,very thick, darkgreen, dashed, -stealth]
  \tikzstyle{r} = [draw, very thick, Red,-stealth,bend left=20]
  %\tikzstyle{r2} = [draw, very thick, Red]
  \tikzstyle{p} = [draw, very thick, dashed, amethyst,-stealth]
  \[
    \begin{tikzpicture}[scale=1]
      \node (0) at (135:1.5) [v] {{\small $0$}};
      \node (1) at (45:1.5) [v] {\small {$a$}};
      \node (3) at (-135:1.5) [v] {{\small $c$}};
      \node (2) at (-45:1.5) [v] {{\small $b$}};   
      \draw [bb] (0) to (1);
      \draw [bb] (2) to (3);
      \draw [rr] (0) to (3);
      \draw [rr] (1) to (2);
      \draw [g-dashed,bend left=20] (3) to (1);
      \draw [g-dashed,bend left=40] (1) to (2);
      \draw [g-dashed,bend left=40] (2) to (3);
    \begin{scope}[shift={(5.5,0)}]
      \node (0) at (90:1.5) [v] {{\small $0$}};
      \node (1) at (18:1.5) [v] {\small {$1$}};
      \node (2) at (-54:1.5) [v] {{\small $2$}};
      \node (3) at (234:1.5) [v] {{\small $3$}};
      \node (4) at (162:1.5) [v] {{\small $4$}};
      \draw [r] (0) to (1);
      \draw [r] (1) to (2);
      \draw [r] (2) to (3);
      \draw [r] (3) to (4);
      \draw [r] (4) to (0);
      \draw [g-dashed,bend right=20] (1) to (2);
      \draw [g-dashed,bend right=20] (2) to (4);
      \draw [g-dashed,bend left=20] (4) to (3);
      \draw [g-dashed,bend left=20] (3) to (1);
    \end{scope}
    \begin{scope}[shift={(10,-1.25)}]
      \node (2) at (0,0) [v] {\footnotesize $b$};
      \node (0) at (-0.1,2) [v] {\footnotesize $0$};
      \node (3) at (2,-.4) [v] {\footnotesize $c$};
      \node (1) at (2.1,1.7) [v] {\footnotesize $a$};
      \node (6) at (1,1.1) [v] {\footnotesize $f$};
      \node (4) at (.9,3) [v] {\footnotesize $d$};
      \node (7) at (2.7,.8) [v] {\footnotesize $g$};
      \node (5) at (2.8,2.8) [v] {\footnotesize $e$}; 
      \draw [rr] (0) to (1);
      \draw [rr] (2) to (3);
      \draw [bb] (0) to (4);
      \draw [bb] (1) to (5);
      \draw [bb] (2) to (6);
      \draw [bb] (3) to (7);
      \draw [g] (0) to (2);
      \draw [g] (1) to (3);
      \draw [rr] (4) to (5);
      \draw [rr] (6) to (7);
      \draw [g] (4) to (6);
      \draw [g] (5) to (7);
      \draw [p,bend left=20] (1) to (2);
      \draw [p,bend left=10] (2) to (4);
      \draw [p] (4) to (3);
      \draw [p,bend left=35] (3) to (6);
      \draw [p,bend right=40] (6) to (7);
      \draw [p,bend right=40] (7) to (5);
      \draw [p,bend right=55] (5) to (1);
    \end{scope}
    \end{tikzpicture}
    \]
    \begin{enumerate}
    \item Create Cayley tables for these fields, for both addition and
      multiplication.
    \item Create Cayley diagrams for the finite fields $\F_3$, $\F_7$,
      and $\F_{11}$.
    \end{enumerate}
  
  %%------------------------------------------------------------------
  
\item Let $I$ and $J$ be ideals of a ring $R$.
  \begin{enumerate}
  \item Show that if a left ideal $I$ of a ring $R$ contains a unit,
  then $I=R$.  
\item Show that $I+J$, $I\cap J$, and $IJ$ are ideals of $R$, where
  \[
  IJ=\big\{x_1y_1+\cdots+x_ky_k\mid x_i\in I,\;y_j\in J\big\}.
  \]
  \item If $R$ is commutative, then the set
    \[
    (I:J)=\big\{r\in R\mid rJ\subseteq I\big\}
    \]
    is called the \emph{colon ideal} of $I$ and $J$. Show that $(I:J)$
    is an ideal of $R$.
  \item Consider the ideals $I=4\Z$ and $J=6\Z$ of the ring
    $R=\Z$. Compute $I+J$, $I\cap J$, $IJ$, $(I:J)$, and $(J:I)$.
  \item Repeat Part~(c) for the ideals $I=m\Z$ and $J=n\Z$ of $R=\Z$. \bigskip 
  \end{enumerate}
  
  %%------------------------------------------------------------------
    


\item The left ideal generated by $X\subseteq R$ is defined as 
 \[
  (X):=\bigcap\;\big\{I\mid\text{$I$ is\ a
   left ideal s.t. $X\subseteq I\subseteq R$}\big\}.
  \]
\begin{enumerate}
  \item Show that the left ideal generated by $X$ is 
    \[
    (X)=\big\{r_1x_1+\cdots+r_nx_n\mid n\in\N,\;r_i\in R,\;x_i\in X\big\}.
    \]
  \item The two-sided ideal generated by $X\subseteq R$ is defined by
    relacing ``left'' with ``two-sided'' in the definition
    above. Show that this is also equal to
    \[
    \big\{r_1x_1s_1+\cdots+r_nx_ns_n\mid n\in\N,\;r_i,s_i\in R,\;x_i\in X\big\}.
    \]
    \item Find a (non-commutive) ring $R$ and a set $X$ such that the left
    and two-sided ideals generated by $X$ are different. \bigskip
\end{enumerate}

  
  
\item Prove the Fundamental homomorphism theorem (FHT) for rings: If
  $\phi\colon R\to S$ is a ring homomorphism, then $\Ker(\phi)$ is a
  two-sided ideal of $R$, and $R/\Ker(\phi)\cong\Image(\phi)$. You may
  assume the FHT for groups.

  
  
\end{enumerate}

\end{document}



