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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 7.1  \\ Fall 2019 \hfill Due October 30}}}

\large

\vspace{1cm}

\begin{exercise}
 \begin{itemize}
 
 \item[\bf a.]  Show that $X+1$ is a unit in $\Z[[X]]$, but is not a unit in $\Z[X]$.
 
 \item[\bf b.]  Show that $X^2 + 3X + 2$ is irreducible in $\Z[[X]]$, but is not irreducible in $\Z[X]$.  
 
 \end{itemize}
\end{exercise}

\begin{exercise}
Let $R$ be a commutative ring and $S \subset R$ a proper multiplicative subset.  

\begin{itemize}

\item[\bf a.]  Prove that for all $s \in S$ and $a \in R$, if $sa=0$, then $a=0$.  

\item[\bf b.]  Show that for all $\frac{a}{s}, \frac{b}{t} \in S^{-1}R$ one has $\frac{a}{s} = \frac{b}{t}$ if and only if $at - bs = 0$.  

\end{itemize}
 
\end{exercise}

\begin{exercise}
Let $S$ be a multiplicative set in a commutative ring $R$.  Suppose $s \in S$, $a \in R$ and $a \ne 0$.  Prove that if $as = 0$, then $\frac{a}{1} = \frac{0}{1}$.  Conclude that if $0 \in S$, then $S^{-1} R = \left\{ \frac{0}{1} \right\}$.
\end{exercise}

\begin{exercise}
If $S \subset \Z$ is a proper multiplicative set, then we can view $S^{-1} \Z$ as a subset of $\Q$.  

\begin{itemize}

\item[\bf a.]  Show that if $S$ is the set of even integers, then $S^{-1} \Z = \Q$.

\item[\bf b.]  State and prove a condition on (a general) $S$ that guarantees $S^{-1} \Z = \Q$.

\end{itemize}

\end{exercise}





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