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\begin{center}
  {\large Mathematics 111}\\Spring 2007\\Homework 2
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\begin{enumerate}
\item Let $A$ be a ring with identity and consider the short exact
  sequence of $A$-modules:
\[\xymatrix{%
  0 \ar@{>}[r]& M' \ar@{>}[r]^\varphi &M \ar@{>}[r]^\psi&M''\ar@{>}[r]&0}
\]
\begin{enumerate}
\item Show that if $M'$ and $M''$ are finitely generated, so is $M$.
\item Show that if $M$ is finitely generated, so is $M''$.
\item Show by example that if $M$ is finitely generated, $M'$ need not be.
\end{enumerate}


\item Problem 5, page 166 of Lang:  Let $A$ be an additive subgroup of
  $\R^n$ (i.e. a $\Z$-module).  Suppose that for any bounded subset
  $B$ of $\R^n$, $A \cap B$ is finite.  Show that $A$ is a free
  $\Z$-module of rank $m \le n$.

Following Artin and Whaple's original proof, Lang gives a detailed
hint.  Make sure you work out the base case carefully and the
inductive step will be easier.


\end{enumerate}







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