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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due November 19 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ 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 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 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 $\Z_2[x]/(x^2+x+1)\cong\F_4$.
  %%
  \[
  \colorlet{lightgrey}{black!15}
  \colorlet{lred}{red!70!white}
  \colorlet{lblue}{blue!65!white}
  \colorlet{lgreen}{green!70!black}
  \colorlet{lyellow}{yellow!60!white}
  \renewcommand*{\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=0.85]
      %%
      \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={(7,0)}]     
        \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={(14,1)}]
        \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=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}
  \]
    The polynomials $f(x)=x^3+x+1\in\Z_2[x]$ and
    $g(x)=x^2+x+2\in\Z_3[x]$ are irreducible. Construct the Cayley
    tables and Cayley diagram for the finite fields
      \[
      \F_8\cong \Z_2[x]/(f)\,\qquad\text{and}\qquad
      \F_9\cong \Z_3[x]/(g)\,.
      \]
      What familiar groups appear as the additive and multiplicative
      groups of these fields? \bigskip
  

\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}



