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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 9.1  \\ Fall 2019 \hfill Due November 13}}}

\large

\vspace{1cm}

\begin{exercise}
Let $R$ be a domain and let $\pi \in R$.  Prove that if $\pi$ is irreducible in $R$, then it is irreducible in $R[X]$.
\end{exercise}


\begin{exercise}
Prove that every GCD domain is an AP domain.  [{\em Suggestion:}  You may assume the following nontrivial fact.  If $R$ is a GCD domain and $a,b ,c \in R$, then $\gcd(a,b) = \gcd(a,c)=R^{\times}$ implies $\gcd(a,bc) =R^{\times}$.]
\end{exercise}

\begin{exercise}
Let $R$ be a domain.  Prove that if every {\em pair} of elements in $R$ has a GCD, then $R$ is a GCD domain.
\end{exercise}


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