\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{enumerate}
\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}

\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{\Image}{Im}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\id}{id}

\def\<{\langle}
\def\>{\rangle}
\def\into{\hookrightarrow}
\def\longto{\longrightarrow}
\def\normal{\vartriangleleft}
\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 June 17 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer I 2019 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

\item For each of the following rings $R$, determine the zero divisors
  (right and left, if appropriate), 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. \medskip
 \end{enumerate}


%\item Let $R$ be an integral domain and $0\neq a\in R$. Prove that
%  $a^k\neq 0$ for all $k\in\N$.


\item Prove that if a left ideal $I$ of a ring $R$ contains a unit,
  then $I=R$. \medskip



\item Let  $I$ and $J$ be ideals of a ring $R$.
\begin{enumerate}
\item Prove that $I+J$, $I\cap J$, and $IJ$ are ideals of $R$. 
\item If $R$ is commutative, then thet set $(I:J)=\{r\in R\mid
  rJ\subset I\}$ is called the \emph{ideal quotient} or \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 $R=\Z$. \medskip
\end{enumerate}




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



\item The finite field $\F_4$ on $4$ elements can be constructed as
  the quotient of the polynomial $\Z_2[x]$ by the ideal $I=(x^2+x+1)$
  generated by the irreducible polynomial $x^2+x+1$. The figure below
  shows a Cayley diagram, and multiplication and addition tables for
  the finite field $\Z_2[x]/(x^2+x+1)\cong\F_4$.
  %%
  \begin{center}
    \colorlet{lightgrey}{black!15}
    \colorlet{lred}{red!70!white}
    \colorlet{lblue}{blue!65!white}
    \colorlet{lgreen}{green!70!black}
    \colorlet{lyellow}{yellow!60!white}
    \newcommand*{\n}{5}
    %%    
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt,
      minimum size=7.5mm] 
    \tikzstyle{g-v} = [circle, draw, fill=darkgreen,inner sep=0pt,
      minimum size=6mm] 
    \tikzstyle{b} = [draw,very thick, Blue]
    \tikzstyle{g-dashed} = [draw,very thick, lgreen, dashed, -stealth]
    \tikzstyle{r} = [draw, very thick, Red,-stealth,bend left=20]
    \tikzstyle{r2} = [draw, very thick, Red]
    \tikzstyle{p} = [draw, very thick, dashed, darkpurple,-stealth]

    \begin{tikzpicture}[scale=0.75]
      %%
      \begin{scope}[shift={(2,3.5)}]     
        \node (0) at (135:2.5) [v] {{\small $0$}};
        \node (1) at (45:2.5) [v] {\small {$1$}};
        \node (3) at (-135:2.5) [v] {{\small $x$}};
        \node (2) at (-45:2.5) [v] {{\small $x\!+\!1$}};   
        \draw [r2] (0) to (1);
        \draw [r2] (2) to (3);
        \draw [b] (0) to (3);
        \draw [b] (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);
      \end{scope}
      %%
      \begin{scope}[shift={(8,0)}]     
        \path[fill=lyellow] (-.1,4) rectangle ++(1,1);
        \path[fill=lred] (-.1,3) rectangle ++(1,1);
        \path[fill=lblue] (-.1,2) rectangle ++(1,1);
        \path[fill=lgreen] (-.1,1) rectangle ++(1,1);
        %%
        \path[fill=lyellow] (1,5.1) rectangle ++(1,.9);
        \path[fill=lred] (2,5.1) rectangle ++(1,.9);
        \path[fill=lblue] (3,5.1) rectangle ++(1,.9);
        \path[fill=lgreen] (4,5.1) rectangle ++(1,.9);
        %%
        \path[fill=lyellow] (1,4) rectangle ++(1,1);
        \path[fill=lyellow] (2,3) rectangle ++(1,1);
        \path[fill=lyellow] (3,2) rectangle ++(1,1);
        \path[fill=lyellow] (4,1) rectangle ++(1,1);
        %%
        \path[fill=lred] (1,3) rectangle ++(1,1);
        \path[fill=lred] (2,4) rectangle ++(1,1);
        \path[fill=lred] (3,1) rectangle ++(1,1);
        \path[fill=lred] (4,2) rectangle ++(1,1);
        %%
        \path[fill=lblue] (1,2) rectangle ++(1,1);
        \path[fill=lblue] (2,1) rectangle ++(1,1);
        \path[fill=lblue] (3,4) rectangle ++(1,1);
        \path[fill=lblue] (4,3) rectangle ++(1,1);
        %%
        \path[fill=lgreen] (1,1) rectangle ++(1,1);
        \path[fill=lgreen] (2,2) rectangle ++(1,1);
        \path[fill=lgreen] (3,3) rectangle ++(1,1);
        \path[fill=lgreen] (4,4) rectangle ++(1,1);
        %%
        \foreach \i in {1,...,\n} {
          \draw [very thin] (\i,1) -- (\i,\n); 
          \draw [very thin] (\i,\n+.1) -- (\i,\n+1); 
          \draw [very thin] (1,\i) -- (\n,\i); 
          \draw [very thin] (-.1,\i) -- (.9,\i); 
        } 
        \draw [very thin] (1,\n+.1) rectangle (\n,\n+1);
        \draw [very thin] (-.1,1) rectangle (.9,\n); 
        \node at (0.4,5.6) {\Large $+$};
        \node at (0.4,4.5) {\footnotesize $0$}; 
        \node at (0.4,3.5) {\footnotesize $1$};
        \node at (0.4,2.5) {\footnotesize $x$};
        \node at (0.4,1.5) {\footnotesize $x\!+\!1$};
        \node at (1.5,5.5) {\footnotesize $0$}; 
        \node at (2.5,5.5) {\footnotesize $1$};
        \node at (3.5,5.5) {\footnotesize $x$};
        \node at (4.5,5.5) {\footnotesize $x\!+\!1$};
        %%
        \node at (1.5,4.5) {\footnotesize $0$}; 
        \node at (1.5,3.5) {\footnotesize $1$};
        \node at (1.5,2.5) {\footnotesize $x$};
        \node at (1.5,1.5) {\footnotesize $x\!+\!1$};
        %%
        \node at (2.5,4.5) {\footnotesize $1$}; 
        \node at (2.5,3.5) {\footnotesize $0$};
        \node at (2.5,2.5) {\footnotesize $x\!+\!1$};
        \node at (2.5,1.5) {\footnotesize $x$};
        %%
        \node at (3.5,4.5) {\footnotesize $x$}; 
        \node at (3.5,3.5) {\footnotesize $x\!+\!1$};
        \node at (3.5,2.5) {\footnotesize $0$};
        \node at (3.5,1.5) {\footnotesize $1$};
        %%
        \node at (4.5,4.5) {\footnotesize $x\!+\!1$}; 
        \node at (4.5,3.5) {\footnotesize $x$};
        \node at (4.5,2.5) {\footnotesize $1$};
        \node at (4.5,1.5) {\footnotesize $0$};
        %%
      \end{scope}
      %%
      \begin{scope}[shift={(16,0)}]
        \path[fill=lred] (-.1,3) rectangle ++(1,1);
        \path[fill=lblue] (-.1,2) rectangle ++(1,1);
        \path[fill=lgreen] (-.1,1) rectangle ++(1,1);
        %%
        \path[fill=lred] (1,4.1) rectangle ++(1,.9);
        \path[fill=lblue] (2,4.1) rectangle ++(1,.9);
        \path[fill=lgreen] (3,4.1) rectangle ++(1,.9);
        %%
        \path[fill=lgreen] (2,2) rectangle ++(1,1);
        \path[fill=lgreen] (3,3) rectangle ++(1,1);
        %%
        \path[fill=lred] (1,3) rectangle ++(1,1);
        \path[fill=lblue] (3,1) rectangle ++(1,1);
        %%
        \path[fill=lblue] (1,2) rectangle ++(1,1);
        \path[fill=lred] (2,1) rectangle ++(1,1);
        %%
        \path[fill=lgreen] (1,1) rectangle ++(1,1);
        \path[fill=lblue] (2,3) rectangle ++(1,1);
        \path[fill=lred] (3,2) rectangle ++(1,1);
         %%
        \foreach \i in {1,2,3,4} {
          \draw [very thin] (\i,1) -- (\i,\n-1); 
          \draw [very thin] (\i,\n-1+.1) -- (\i,\n); 
          \draw [very thin] (1,\i) -- (\n-1,\i); 
          \draw [very thin] (-.1,\i) -- (.9,\i); 
        } 
        \draw [very thin] (1,\n-1+.1) rectangle (\n-1,\n);
        \draw [very thin] (-.1,1) rectangle (.9,\n-1); 
        \node at (0.4,4.6) {\Large $\times$};
        \node at (0.4,3.5) {\footnotesize $1$};
        \node at (0.4,2.5) {\footnotesize $x$};
        \node at (0.4,1.5) {\footnotesize $x\!+\!1$};
        \node at (1.5,4.5) {\footnotesize $1$}; 
        \node at (2.5,4.5) {\footnotesize $x$};
        \node at (3.5,4.5) {\footnotesize $x\!+\!1$};
        %%
        \node at (1.5,3.5) {\footnotesize $1$};
        \node at (1.5,2.5) {\footnotesize $x$};
        \node at (1.5,1.5) {\footnotesize $x\!+\!1$};
        %%
        \node at (2.5,3.5) {\footnotesize $x$};
        \node at (2.5,2.5) {\footnotesize $x\!+\!1$};
        \node at (2.5,1.5) {\footnotesize $1$};
        %%
        \node at (3.5,3.5) {\footnotesize $x\!+\!1$};
        \node at (3.5,2.5) {\footnotesize $1$};
        \node at (3.5,1.5) {\footnotesize $x$};
        %%
        \end{scope}
        %%
    \end{tikzpicture}
    \end{center}
  %%
  \begin{enumerate}
    \item Find a degree-$3$ polynomial $f\in\Z_2[x]$ that is
      irreducible over $\Z_2$, and a degree-$2$ polynomial
      $g\in\Z_3[x]$ that is irreducible over $\Z_3$. [\emph{Hint}:
        Any polynomial with no roots in the ``prime field'' $\Z_p$ will work.]
    \item Construct Cayley diagrams, addition, and multiplication
      tables for the finite fields
      \[
      \F_8\cong \Z_2[x]/(f)\,\qquad\text{and}\qquad
      \F_9\cong \Z_3[x]/(g)\,.
      \] 
  \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}

