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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 1.2  \\ Fall 2019 \hfill Due September 4}}}

\large

\vspace{1cm}

\begin{exercise}
Let $p \in \N$ be a prime and let $\Z_{(p)}$ denote the set of all rational numbers with denominators {\em not} divisible by $p$.    

\begin{itemize}

\item[\bf a.]  Show that $\Z_{(p)}$ is a subring of $\Q$.

\item[\bf b.]  Determine $\Z_{(p)}^{\times}$.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $\mathbb{H}$  be the set of all $2 \times 2$ matrices over $\C$ of the form
\[
\left( \begin{matrix} z & w \\ -\overline{w} & \overline{z} \end{matrix} \right),
\]
where the bar denotes complex conjugation.  Prove that $\mathbb{H}$ is a noncommutative division ring.\footnote{A {\em division ring} (or {\em skew field}) is a ring (not necessarily commutative) in which every nonzero element is a unit.}

\end{exercise}

\begin{exercise}
Let $R$ be a ring and $a, b, c \in R$.  Prove the following {\em left cancellation law}:  if $a \ne 0$ and $a$ is not a zero divisor, then $ab = ac$ implies $b = c$.


\end{exercise}

\begin{exercise}
Prove that if $F$ is a field, then every nonzero element of $\operatorname{M}_n(F)$ is either a unit or a zero divisor.  Show that this is no longer true if $F$ is replaced with $\Z$.
\end{exercise}

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