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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 7.2  \\ Fall 2019 \hfill Due October 30}}}

\large

\vspace{1cm}

\begin{exercise}
Let $S$ be a nonempty subset of a commutative ring $R$.  An element $d \in R$ is a {\em greatest common divisor} (GCD) of $S$ provided:
\begin{itemize}

\item[\bf (i)]  $d | a$ for all $a \in S$;

\item[\bf (ii)]  $c \in R$ and $c|a$ for all $a \in S$ implies $c|d$.

\end{itemize}

\noindent Prove the following.

\begin{itemize}

\item[\bf a.]  $S$ has GCD $d$ if and only if $\{ (c) \, | \, (S) \subset (c) \}$ has least element $(d)$ (under inclusion).

\item[\bf b.]  Any two GCDs of $S$ are associate.

\end{itemize}
\end{exercise}

\begin{exercise}
A {\em B{\'e}zout domain} is a domain in which every finitely generated ideal is principal.  A {\em GCD domain} is a domain with the property that every nonempty finite subset has a GCD.  Prove the following.

\begin{itemize}

\item[\bf a.]  Every PID is a GCD domain.

\item[\bf b.]  Every B{\'e}zout domain is a GCD domain.

\item[\bf c.]  Every Noetherian B{\'e}zout domain is a PID.

\item[\bf d.]  Every UFD is a GCD domain.

\end{itemize}

\noindent[{\em Suggestions:}  Use Exercise {\bf 1a} for parts {\bf a} - {\bf c}.  For part {\bf d}, given finitely many elements in a UFD, use their prime factorizations to construct a GCD.]


\end{exercise}


\begin{exercise}
 Prove that the image or inverse image of a multiplicative set under a ring homomorphism is also a multiplicative set.
\end{exercise}

\begin{exercise}
Prove that the intersection of any collection of multiplicative sets in a ring $R$ is again a multiplicative set.  In particular, the complement of a union of prime ideals in $R$ is a multiplicative set.  
\end{exercise}



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