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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due November 12 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}


\begin{enumerate}
  
\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 the 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 $R=\Z$. \\
  \end{enumerate}
  
  

\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 If $F$ is a field, show that the ring $M_n(F)$ of $n\times n$
  matrices over $F$ is simple. \\
  

  
\item Let $p$ be a fixed prime number, and consider the ring
  \[
  R=\big\{\tfrac{a}{b}\mid a,b\in\Z,\;(a,b)=1,\; p\nmid b\big\}
  \]
  with the usual operations of addition and multiplication of rational
  numbers.
  \begin{enumerate} 
  \item Determine the group of units of $R$.
  \item Prove that the principal ideal $(p)=pR$ is a maximal ideal of
    $R$, and in fact, the only maximal ideal of $R$. \\
  \end{enumerate}
  
  
  
\item An element $a$ of a ring $R$ is called \emph{nilpotent} if
  $a^n=0$ for some positive integer $n$.
  \begin{enumerate}
  \item Show that the set of nilpotent elements in a commutative ring
    $R$ forms an ideal.
  \item If $u\in R$ is a unit and $a\in R$ nilpotent, show that $u+a$ is
    a unit.
  \item Find $U(R)$ if $R=\Z_{p^k}[x]$, where $p$ is prime and $k\geq
    1$. Prove all of your claims. \\
  \end{enumerate}

\end{enumerate}

\end{document}
