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\title{Math 332: Homework 11}
%\author{Your name here --- delete the % at the start of the line}
\date{27.IV.15}

\begin{document}
\maketitle

\begin{exc}
Compute
\[
  \Hom_\ZZ(\ZZ/65\ZZ\otimes_\ZZ \ZZ/91\ZZ, \ZZ/131\ZZ)
\]
as an abelian group.
\end{exc}

\begin{exc}
Find, with proof, the number of finitely generated abelian groups of order $100$.  Do the same for finitely generated abelian groups of order $576$.
\end{exc}

\begin{prob}
Suppose $R$ is a commutative ring and that $M$ and $N$ are free $R$-modules of ranks $m$ and $n$, respectively.  Show that $M\otimes_R N$ is a free $R$-module of rank $mn$.  Find a basis for $M\otimes_R N$ in terms of bases $x_1,\ldots,x_m$ and $y_1,\ldots,y_n$ of $M$ and $N$, respectively.
\end{prob}

\begin{prob}
Let $M$ be a module over an integral domain $R$.
\begin{enumerate}[(a)]
\item Suppose $x$ is a nonzero torsion element in $M$.  Show that $x$ and $0$ are linearly dependent.  Conclude that the rank of $\Tor(M)$ is $0$, so in particular, any torsion $R$-module has rank $0$.
\item Show that the rank of $M$ is the same as the rank of $M/\Tor(M)$.
\item If $R$ is a PID and $M$ is a finitely generated $R$-module, describe the structure of $M/\Tor(M)$.
\end{enumerate}
\end{prob}

\begin{prob}
Let $R$ be a PID and let $M$ be a finitely generated $R$-module.  By the structure theorem,
\[
  M\cong R^r \oplus R/(a_1)\oplus\cdots\oplus R/(a_d)
\]
for some $r,d\ge 0$ and $a_i$ nonzero, nonunit elements of $R$ such that $a_1\mid a_2\mid \cdots\mid a_d$.  As such, there is a canonical map $R^{r+d}\to M$.  Describe this map and its kernel; in particular, prove that the kernel is a free $R$-module.
\end{prob}

\begin{prob}
Let $G$ be the
 quotient group $\QQ/\ZZ$.  Is $G$ a free $\ZZ$-module? Torsion-free? Finitely-generated?
\end{prob}

\begin{chal}
Let $A$ be a subgroup of $\RR^n$ such that in each ball in  $\RR^n$ there are only finitely many elements of $A$.  Show that $A$ is a free abelian group on at most $n$ generators.
\end{chal}

\begin{chal}
Find a commutative ring $R$ and finitely generated $R$-module $M$ such that there is no $R$-module homomorphism from a finite rank free $R$-module to $M$ with free kernel.  (Compare with the situation in Problem 5.)
\end{chal}

\end{document}