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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due November 14 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}


%%---------------------------------------------------------------------  
  
\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 $R=\Z_6\times
    \Z_4$, whose subring lattice is shown below.
  \item Describe how to find $IJ$ and $(I:J)$ by inspection,
    using only the subring lattice, if possible. \vspace{-8mm}
  \end{enumerate}

  \[
  \hspace*{-8mm}
   \newcommand\Hh{6-.1} % height of G
   \newcommand\Hg{4.75-.1} % height of C_{12},  C_6 x C_2 
   \newcommand\Hf{3.75} % height of C_4 x C2  
   \newcommand\He{3.5-.1} % height of C_6  
   \newcommand\Hd{2.5} % height of C_4, V_4
   \newcommand\Hc{2.2-.1} % height of C_3  
   \newcommand\Hb{1.25} % height of C_2 
   \newcommand\Ha{0} % height of C_1  
  \colorlet{Ideal}{Blue}
  \colorlet{Subring}{Red}
  \colorlet{Neither}{faded}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.4]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(7,1.3)},xscale=1.5]
      \node[anchor=east] at (-3,\Hh) {\normalsize \textbf{Order} $=24$};
      \node[anchor=east]  at (-3,\Hg) {\normalsize $12$};
      \node[anchor=east]  at (-3,\Hf) {\normalsize $8$};
      \node[anchor=east]  at (-3,\He) {\normalsize $6$};
      \node[anchor=east]  at (-3,\Hd) {\normalsize $4$};
      \node[anchor=east]  at (-3,\Hc) {\normalsize $3$};
      \node[anchor=east]  at (-3,\Hb) {\normalsize $2$};
      \node[anchor=east]  at (-3,\Ha) {\normalsize $1$};
      %%
      \node[anchor=east] at (3.5,\Hh) {\normalsize \textbf{Index} $=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[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\>$}; % C_4
      \node[Ideal] (01) at (2.5,\Hd) {$\<01\>$}; % C_4
      %%      
      \node[Ideal] (20) at (-1.25,\Hc) {$\<20\>$};  % C_3
      %%
      \node[Ideal] (30) at (-1.25,\Hb) {$\<30\>$};  % C_2 
      \node[Subring] (32) at (0,\Hb) {$\<32\>$};  % C_2 
      \node[Ideal] (02) at (1.25,\Hb) {$\<02\>$};  % C_2 
      %%
      \node[Ideal] (00) at (0,\Ha) {$\<00\>$}; % C_1
      %%
      \draw[faded] (G) to (10-02); \draw[faded] (G) to (21);  
      \draw[faded] (G) to (11);  \draw[faded] (G) to (30-01); 
      %%
      \draw[faded] (10-02) to (30-02); \draw[faded] (10-02) to (12);
      \draw[faded] (10-02) to (22); \draw[faded] (10-02) to (10);
      \draw[faded] (30-01) to (01); \draw[faded] (30-01) to (31); 
      \draw[faded] (30-01) to (30-02);
      \draw[faded] (10) to (20); \draw[faded] (10) to (30);
      \draw[faded] (22) to (20); \draw[faded] (22) to (02);
      \draw[faded] (12) to (20); \draw[faded] (12) to (32);
      \draw[faded] (01) to (02); \draw[faded] (31) to (02);
      \draw[faded] (21) to (22); \draw[faded] (21) to (01);
      \draw[faded] (11) to (22); \draw[faded] (11) to (31);
      %%
      \draw[faded] (30-02) to (30); \draw[faded] (30-02) to (02); 
      \draw[faded] (30-02) to (32);
      \draw[faded] (00) to (02); \draw[faded] (00) to (32); \draw[faded] (00) to (30); \draw[faded] (00) to (20); 
    \end{scope}
  \end{tikzpicture}
\]
 
%%---------------------------------------------------------------------
  
\item Let $f:R\rightarrow S$ 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.
  \end{enumerate}
  
%%---------------------------------------------------------------------
  
\item Let $R$ be a commutative ring.
  \begin{enumerate}
    \item Show that if $x$ is contained in every maximal ideal, then
      $1+x$ is a unit.
    \item A ring is \emph{local} if it has a unique maximal
      ideal. Show that $R$ is local if and only if the non-units form
      an ideal.
    \item Characterize units and maximal ideals of the ring
      \[
      R=\big\{\tfrac{a}{b}\mid a,b\in\Z,\;(a,b)=1,\; p\nmid b\big\}\subseteq\Q,
      \]
      where $p$ is a fixed prime.
  \end{enumerate}

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

\item Let $R$ be a commutative ring. In this problem, we will define
  several different ``radicals'' of an ideal $I$.
  \begin{enumerate}
  \item The \emph{radical} of $I\subseteq R$ is the set
    \[
    \sqrt{I}:=\big\{x\in R\mid x^n\in I\text{ for some } n\in\N\big\},
    \]
    and $I$ is a \emph{radical ideal} if $\sqrt{I}=I$.
    \begin{enumerate}[(i)]
    \item Show that $\sqrt{I}$ is an ideal containing $I$.
    \item Find the radicals of all ideals of the rings
      $\Z_6\times\Z_4$, $\Z_8\times\Z_2$, $\Z_6\times\Z_2$, and
      $\Z_{24}$. Denote these on the subring lattices by drawing an
      arrow from each $I$ to $\sqrt{I}$.
    \end{enumerate}
    %%
  \item The \emph{Jacobsen radical} of $I$, denoted $J(I)$, is the
    intersection of all maximal ideals that contain $I$. 
    \begin{enumerate}[(i)]
    \item Show that $J(I)$ is an ideal.
    \item Find the Jacobsen radical of all proper ideals of the rings
      $\Z_{24}$, $\Z_6\times\Z_4$, $\Z_6\times\Z_2$, and
      $\Z_8\times\Z_2$. Denote these by drawing an arrow from $I$ to
      $J(I)$ on a fresh copy of the lattices.  \vspace{-8mm}
    \end{enumerate}
  \end{enumerate}
 
  \[
  \hspace*{-10mm}
   \newcommand\Hg{6} % height of C_{12},  C_6 x C_2 
   \newcommand\He{5} % height of C_6  
   \newcommand\Hd{4} % height of C_4, V_4
   \newcommand\Hc{3} % height of C_3  
   \newcommand\Hb{2} % height of C_2 
   \newcommand\Ha{1} % height of C_1  
  \colorlet{Ideal}{Blue}
  \colorlet{Subring}{Red}
  \colorlet{Neither}{faded}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(-6.5,.3)},scale=.91,xscale=.7]
      \node[Ideal] (G) at (3.5,6+.3) {$\Z_{24}$};
      \node[Ideal] (r2-s) at (1.75,4.5+.3) {$\<2\>$};
      \node[Ideal] (r) at (5.25,4.5-.3) {$\<3\>$};
      \node[Ideal] (r4-s) at (0,3+.3) {$\<4\>$};
      \node[Ideal] (r2) at (3.5,3-.3) {$\<6\>$};
      \node[Ideal] (s) at (-1.75,1.5+.3) {$\<8\>$};
      \node[Ideal] (r4) at (1.75,1.5-.3) {$\<12\>$};
      \node[Ideal] (1) at (0,0-.3) {$\<0\>$};
      \draw[faded] (1) to (s);
      \draw[faded] (1) to (r4);
      \draw[faded] (s) to (r4-s);
      \draw[faded] (r4) to (r4-s);
      \draw[faded] (r4) to (r2);
      \draw[faded] (r4-s) to (r2-s);
      \draw[faded] (r2) to (r2-s);
      \draw[faded] (r2) to (r);
      \draw[faded] (r2-s) to (G);
      \draw[faded] (r) to (G);
    \end{scope}
    %%
    \begin{scope}[shift={(0,0)},scale=1.2,xscale=.8]
      \node[Ideal] (G) at (0,5) {$\Z_6\times\Z_2$};
      \node[Ideal] (21) at (1,4) {$\<21\>$};
      \node[Subring] (11) at (2,4) {$\<11\>$};
      \node[Ideal] (10) at (3,4) {$\<10\>$};
      \node[Ideal] (01-30) at (-2,3) {$\<01,30\>$};
      \node[Ideal] (20) at (2,2) {$\<20\>$};
      \node[Ideal] (01) at (-3,1) {$\<01\>$};
      \node[Subring] (31) at (-2,1) {$\<31\>$};
      \node[Ideal] (30) at (-1,1) {$\<30\>$};
      \node[Ideal] (00) at (0,0) {$\<00\>$};
      \draw[faded] (G) to (21);
      \draw[faded] (G) to (11); 
      \draw[faded] (G) to (10);
      \draw[faded] (20) to (21);
      \draw[faded] (20) to (11); 
      \draw[faded] (20) to (10);
      \draw[faded] (01-30) to (G);
      \draw[faded] (21) to (01); 
      \draw[faded] (11) to (31); 
      \draw[faded] (10) to (30);
      \draw[faded] (00) to (20);
      \draw[faded] (01-30) to (01);
      \draw[faded] (01-30) to (31); 
      \draw[faded] (01-30) to (30); 
      \draw[faded] (00) to (01);
      \draw[faded] (00) to (31); 
      \draw[faded] (00) to (30); 
    \end{scope}
    %%
    \begin{scope}[shift={(4,0)},xscale=.8]
      \tikzstyle{every node}=[font=\footnotesize]
      \node[Ideal] (G) at (3.5,6) {$\Z_8\!\times\!\Z_2$};
      \node[Subring] (rs) at (3.5,4.5) {$\<11\>$};
      \node[Ideal] (r2-s) at (1.75,4.5) {$\<21\>$};
      \node[Ideal] (r) at (5.25,4.5) {$\<10\>$};
      \node[Ideal] (r4-s) at (0,3) {$\<01,40\>$};
      \node[Subring] (r2s) at (1.75,3) {$\<21\>$};
      \node[Ideal] (r2) at (3.5,3) {$\<20\>$};
      \node[Ideal] (s) at (-1.75,1.5) {$\<01\>$};
      \node[Subring] (r4s) at (0,1.5) {$\<41\>$};
      \node[Ideal] (r4) at (1.75,1.5) {$\<40\>$};
      \node[Ideal] (1) at (0,0) {$\<00\>$};
      \draw[faded] (1) -- (s);
      \draw[faded] (1) -- (r4s);
      \draw[faded] (1) -- (r4);
      \draw[faded] (s) -- (r4-s);
      \draw[faded] (r4s) -- (r4-s);
      \draw[faded] (r4) -- (r4-s);
      \draw[faded] (r4) -- (r2s);
      \draw[faded] (r4) -- (r2);
      \draw[faded] (r4-s) -- (r2-s);
      \draw[faded] (r2s) -- (r2-s);
      \draw[faded] (r2) -- (r2-s);
      \draw[faded] (r2) -- (r);
      \draw[faded] (r2) -- (rs);
      \draw[faded] (r2-s) -- (G);
      \draw[faded] (r) -- (G);
      \draw[faded] (rs) -- (G);
    \end{scope}
  \end{tikzpicture}
  \]

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

\item Let $R$ be commutative. Loosely speaking, a \emph{radical} of
  $R$ is an ideal of ``bad elements.'' \vspace{-1mm}
  \begin{enumerate}
  \item An element $a\in R$ is \emph{nilpotent} if $a^n=0$
    for some $n\geq 1$. The \emph{nilradical} of $R$ is
    $\Nil_R:=\sqrt{0}$, the set of nilpotent elements.
    \begin{enumerate}[(i)]
    \item If $u\in R$ is a unit and $a\in R$ is nilpotent, show that $u+a$ is
      a unit.
    \item Show that $R/\Nil_R$ has no nonzero nilpotent elements.
    \item Show that $\Nil_{R/I}=\sqrt{I}/I$.
    \end{enumerate}
  \item The \emph{Jacobsen radical of $R$} is $J(R):=J(0)$, the
    intersection of all maximal ideals. Show that $J(R)\supseteq\Nil_R$. 
  \item Find the Jacobsen and nilradicals of the rings $\Z_{24}$,
    $\Z_6\times\Z_4$, $\Z_6\times\Z_2$, and $\Z_8\times\Z_2$. Beside
    each ideal $I$ in the lattice, write $\Nil_{R/I}$.
  \end{enumerate}


\end{enumerate}

\end{document}
