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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due April 18 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

\item The subgroup lattices of $\Z_4\times \Z_4$ and $\Z_6\times \Z_3$ are shown below. Re-draw these lattices with the subgroups written with generator(s), and then construct their \emph{subring lattices} by coloring each subgroup based on whether it is an \Balert{ideal}, \Alert{subring but not an ideal}, or {\color{faded}subgroup that is not a subring}. \vspace{-2mm}
  \[
  \hspace*{-8mm}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=.9]
    \tikzstyle{every node}=[font=\normalsize]
    \begin{scope}[shift={(0,0)},scale=1.25,yscale=1.25]
      \node (G) at (0,4) {$\Z_4\!\times\!\Z_4$};
      \node (C4xC2-1) at (-1.5,3) {$\Z_4\!\times\!\Z_2$};
      \node (C4xC2-2) at (0,3) {$\Z_4\!\times\!\Z_2$};
      \node (C4xC2-3) at (1.5,3) {$\Z_4\!\times\!\Z_2$};
      \node (C4-1) at (-3,2) {$\Z_4$};
      \node (C4-2) at (-2.25,2) {$\Z_4$};
      \node (C4-3) at (-1.5,2) {$\Z_4$};
      \node (C4-4) at (1.5,2) {$\Z_4$};
      \node (C4-5) at (2.25,2) {$\Z_4$};
      \node (C4-6) at (3,2) {$\Z_4$};
      \node (V4) at (0,2) {$\Z_2^2$};
      \node (C2-1) at (-1.5,1) {$\Z_2$};
      \node (C2-2) at (0,1) {$\Z_2$};
      \node (C2-3) at (1.5,1) {$\Z_2$};
      \node (1) at (0,0) {$\Z_1$};
      \draw[f] (G) to (C4xC2-1);
      \draw[f] (G) to (C4xC2-2);
      \draw[f] (G) to (C4xC2-3);
      \draw[f] (C4xC2-1) to (C4-1);
      \draw[f] (C4xC2-1) to (C4-2);
      \draw[f] (C4xC2-1) to (V4); 
      \draw[f] (C4xC2-2) to (C4-3);
      \draw[f] (C4xC2-2) to (C4-4);
      \draw[f] (C4xC2-2) to (V4); 
      \draw[f] (C4xC2-3) to (C4-5);
      \draw[f] (C4xC2-3) to (C4-6);
      \draw[f] (C4xC2-3) to (V4);
      \draw[f] (C4-1) to (C2-1); \draw[f] (C4-2) to (C2-1);
      \draw[f] (C4-3) to (C2-2); \draw[f] (C4-4) to (C2-2);
      \draw[f] (V4) to (C2-1); 
      \draw[f] (V4) to (C2-2); \draw[f] (V4) to (C2-3);
      \draw[f] (C4-5) to (C2-3); \draw[f] (C4-6) to (C2-3);
      \draw[f] (C2-1) to (1); 
      \draw[f] (C2-2) to (1); 
      \draw[f] (C2-3) to (1);
    \end{scope}
    %%
    \begin{scope}[shift={(9,0)},scale=1.44,yscale=1.5]
      \newcommand\He{3} % order 18
      \newcommand\Hd{2.2} % order 9
     \newcommand\Hc{1.9} % order 6
     \newcommand\Hb{1.1} % order 3
      \newcommand\Ha{.8} % order 2
      %%
      \node (G) at (0,\He) {$\Z_6\!\times\!\Z_3$};
      \node (C3^2) at (-1.5,\Hd) {$\Z_3^2$};
      \node (C6-1) at (-.25,\Hc) {$\Z_6$};
      \node (C6-2) at (.75,\Hc) {$\Z_6$};
      \node (C6-3) at (1.75,\Hc) {$\Z_6$};
      \node (C6-4) at (2.75,\Hc) {$\Z_6$};
      %%
      \node (C3-1) at (-2.75,\Hb) {$\Z_3$};
      \node (C3-2) at (-1.75,\Hb) {$\Z_3$};
      \node (C3-3) at (-.75,\Hb) {$\Z_3$};
      \node (C3-4) at (.25,\Hb) {$\Z_3$};
      \node (C2) at (1.5,\Ha) {$\Z_2$};
      %%
      \node (1) at (0,0) {$\Z_1$};
      %%

      %%
      \draw[f] (G) to (C6-1); \draw[f] (G) to (C6-2); 
      \draw[f] (G) to (C6-3); \draw[f] (G) to (C6-4);
      \draw[f] (G) to (C3^2);
      %%
      \draw[f] (C2) to (C6-1); \draw[f] (C2) to (C6-2);
      \draw[f] (C2) to (C6-3); \draw[f] (C2) to (C6-4);
      %%
      \draw[f] (C6-1) to (C3-1); \draw[f] (C6-2) to (C3-2);
      \draw[f] (C6-3) to (C3-3); \draw[f] (C6-4) to (C3-4);
      %%
      \draw[f] (C3^2) to (C3-1); \draw[f] (C3^2) to (C3-2);
      \draw[f] (C3^2) to (C3-3); \draw[f] (C3^2) to (C3-4);
      %%
      \draw[f] (C3-1) to (1); 
      \draw[f] (C3-2) to (1); 
      \draw[f] (C3-3) to (1);
      \draw[f] (C3-4) to (1);
      \draw[f] (C2) to (1); 
    \end{scope}
    \end{tikzpicture}
    \]
    Finally, for each ring, write down the units, and the zero
    divisors. \medskip
  
 

    %%---------------------------------------------------------------------  
  
\item Let  $I$ and $J$ be ideals of a commutative ring $R$.
  \begin{enumerate}
  \item Show that $I+J$, $I\cap J$, and $IJ$ are ideals of $R$. Which
    of these remain ideals if the commutativity hypothesis is dropped?
  \item The set $(I:J):=\{r\in R\mid rJ\subseteq I\}$ is called the
    \emph{ideal quotient} or \emph{colon ideal} of $I$ and $J$. Show
    that $(I:J)$ is an ideal of $R$. Does this require commutativity?
  \item Determine $I+J$, $I\cap J$, $IJ$, and $(I:J)$ for the ideals
    $I=n\Z$ and $J=m\Z$ of $R=\Z$.
  \item Repeat Part~(c) for several pairs of ideals of $\Z_6\times
    \Z_4$; see the subring lattice below.
  \item Describe how to find $IJ$ and $(I:J)$ by inspection, from the lattice, if possible. \vspace{-3mm}
  \end{enumerate}

  \[
  \hspace*{-8mm}
   \newcommand\Hh{6-.1} % height of G
   \newcommand\Hg{4.75-.1} % height of \Z_{12},  \Z_6 x \Z_2 
   \newcommand\Hf{3.75} % height of \Z_4 x C2  
   \newcommand\He{3.5-.1} % height of \Z_6  
   \newcommand\Hd{2.5} % height of \Z_4, V_4
   \newcommand\Hc{2.2-.1} % height of \Z_3  
   \newcommand\Hb{1.25} % height of \Z_2 
   \newcommand\Ha{0} % height of \Z_1  
  \colorlet{Ideal}{xBlue}
  \colorlet{Subring}{xRed}
  \colorlet{Neither}{faded}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.2]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(7,1.3)},xscale=1.5]
      \node[anchor=east] at (3.5,\Hh) {\normalsize \textbf{Order} $=24$};
      \node[anchor=east]  at (3.5,\Hg) {\normalsize $12$};
      \node[anchor=east]  at (3.5,\Hf) {\normalsize $8$};
      \node[anchor=east]  at (3.5,\He) {\normalsize $6$};
      \node[anchor=east]  at (3.5,\Hd) {\normalsize $4$};
      \node[anchor=east]  at (3.5,\Hc) {\normalsize $3$};
      \node[anchor=east]  at (3.5,\Hb) {\normalsize $2$};
      \node[anchor=east]  at (3.5,\Ha) {\normalsize $1$};
      %%
      \node[anchor=east] at (-3,\Hh) {\normalsize \textbf{Index} $=1$};
      \node[anchor=east]  at (-3,\Hg) {\normalsize $2$};
      \node[anchor=east]  at (-3,\Hf) {\normalsize $3$};
      \node[anchor=east]  at (-3,\He) {\normalsize $4$};
      \node[anchor=east]  at (-3,\Hd) {\normalsize $6$};
      \node[anchor=east]  at (-3,\Hc) {\normalsize $8$};
      \node[anchor=east]  at (-3,\Hb) {\normalsize $12$};
      \node[anchor=east]  at (-3,\Ha) {\normalsize $24$};
      %%
      \node[Ideal] (G) at (0,\Hh) {$\Z_6\times\Z_4$};
      \node[Ideal] (10-02) at (-1.25,\Hg) {$\<10,02\>$}; % Z_6 x Z_2
      \node[Subring] (11) at (0,\Hg) {$\<11\>$}; % Z_{12}
      \node[Ideal] (21) at (1.25,\Hg) {$\<21\>$}; % Z_{12}
      %%
      \node[Ideal] (30-01) at (1.25,\Hf) {$\<30,01\>$}; % Z_2 x Z_4
      %%
      \node[Ideal] (10) at (-2.5,\He) {$\<10\>$};  % Z_6
      \node[Subring] (12) at (-1.25,\He) {$\<12\>$};  % Z_6
      \node[Ideal] (22) at (0,\He) {$\<22\>$};  % Z_6
      \node[Ideal] (30-02) at (0,\Hd) {$\<30,02\>$}; % V_4
      \node[Subring] (31) at (1.25,\Hd) {$\<31\>$}; % \Z_4
      \node[Ideal] (01) at (2.5,\Hd) {$\<01\>$}; % \Z_4
      %%      
      \node[Ideal] (20) at (-1.25,\Hc) {$\<20\>$};  % \Z_3
      %%
      \node[Ideal] (30) at (-1.25,\Hb) {$\<30\>$};  % \Z_2 
      \node[Subring] (32) at (0,\Hb) {$\<32\>$};  % \Z_2 
      \node[Ideal] (02) at (1.25,\Hb) {$\<02\>$};  % \Z_2 
      %%
      \node[Ideal] (00) at (0,\Ha) {$\<00\>$}; % \Z_1
      %%
      \draw[f] (G) to (10-02); \draw[f] (G) to (21);  
      \draw[f] (G) to (11);  \draw[f] (G) to (30-01); 
      %%
      \draw[f] (10-02) to (30-02); \draw[f] (10-02) to (12);
      \draw[f] (10-02) to (22); \draw[f] (10-02) to (10);
      \draw[f] (30-01) to (01); \draw[f] (30-01) to (31); 
      \draw[f] (30-01) to (30-02);
      \draw[f] (10) to (20); \draw[f] (10) to (30);
      \draw[f] (22) to (20); \draw[f] (22) to (02);
      \draw[f] (12) to (20); \draw[f] (12) to (32);
      \draw[f] (01) to (02); \draw[f] (31) to (02);
      \draw[f] (21) to (22); \draw[f] (21) to (01);
      \draw[f] (11) to (22); \draw[f] (11) to (31);
      %%
      \draw[f] (30-02) to (30); \draw[f] (30-02) to (02); 
      \draw[f] (30-02) to (32); \draw[f] (00) to (02);
      \draw[f] (00) to (32); \draw[f] (00) to (30);
      \draw[f] (00) to (20); 
    \end{scope}
  \end{tikzpicture}
\]

%%------------------------------------------------------------------
  
\item Let $f\colon R\rightarrow S$ be a ring homomorphism between
  commutative rings.
  \begin{enumerate} 
  \item If $f$ is surjective and $I$ is an ideal of $R$, show that
    $f(I)$ is an ideal of $S$.
  \item Show that Part~(a) is not true in general when $f$ is not
    surjective.
  \item Show that if $f$ is surjective and $R$ is a field, then $S$ is
    a field as well. \medskip
  \end{enumerate}

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

\end{enumerate}

\end{document} 
    
  
 
