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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 8.1  \\ Fall 2019 \hfill Due November 6}}}

\large

\vspace{1cm}

\begin{exercise}
Let $R$ be a commutative ring, $S \subset R$ be a multiplicative set, and $\varphi_S : R \to S^{-1}R$ the canonical map $a \mapsto \frac{a}{1}$.  Prove that $\vp_S$ is an embedding if and only if $S$ is proper.
\end{exercise}

\begin{exercise}
Let $R$, $S$ and $\vp_S$ be as in the preceding exercise.  Let $\psi :R \to R'$ be a ring homomorphism so that $\psi(S) \subset (R')^{\times}$, and let $\widehat{\psi} : S^{-1}R \to R'$ be the induced map given by $\widehat{\psi} \left( \frac{a}{s} \right) = \psi(a) \psi(s)^{-1}$.  Prove that when $S$ is proper, $\widehat{\psi}$ is injective if and only if $\psi$ is injective.
\end{exercise}

\begin{exercise}
Let $p$ be a prime, $\pi : \Z \to \Z/p\Z$ be the canonical projection and $S = \Z \dashv p \Z$.

\begin{itemize}

\item[\bf a.]  Prove that $\pi(S) \subset (\Z/p\Z)^{\times}$.

\item[\bf b.]  Let $\widehat{\pi} : \Z_{(p)} \to \Z/p\Z$ denote the induced map.  Show that
\[
\ker \widehat{\pi} = p \Z_{(p)}  = \left\{  \frac{a}{s} \, \bigg| \, a , s \in \Z, \, p|a, \, p \nmid s \right\}.
\]

\item[\bf c.]  Conclude that $\Z_{(p)}/p\Z_{(p)} \cong \Z/p\Z$, and hence that $p \Z_{(p)}$ is a maximal ideal in $\Z_{(p)}$.

\end{itemize}

\end{exercise}


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