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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#12}
\date{Assigned Friday, 6 October 2017; due Monday, 9 October 2017}

\maketitle

\begin{problab}{12.1}
Let $M$ be the $\Z$-module $\Z/24\Z \times \Z/15\Z \times \Z/50\Z$.
\begin{enumalph}
\item Find $\ann(M)$, the annihilator of $M$ in $\Z$.
\item Let $I=2\Z$.  Compute the annihilator of $I$ in $M$.
\end{enumalph}
\end{problab}

\vspace{2ex}

Let $R$ be a ring (with $1$).  We say $x \in R$ is a \emph{left zerodivisor} if $x \neq 0$ and there exists a nonzero $y \in R$ such that $xy=0$.  We similarly�define \emph{right zerodivisor}.  If $R$ is commutative, there is no difference between left and right, so we simply say \emph{zerodivisor}.  

We say $R$ is a \emph{domain} if $R$ is commutative and has no zerodivisors.  

\vspace{2ex}

\begin{problab}{12.2} 
Let $R$ be a ring, and let $M$ be left $R$-module.  An element $m \in M$ is called a \emph{torsion element} if $rm=0$ for some nonzero $r \in R$.  The set of torsion elements is denoted $\Tor(M)$.
\begin{enumalph}
\item Prove that if $R$ is a domain, then $\Tor(M)$ is a submodule of $M$.
\item Give an example of a ring $R$ and an $R$-module $M$ such that $\Tor(M)$ is not a submodule.  
\item Show that if $R$ has a zerodivisor then every nonzero $R$-module $M$ has $\Tor(M) \neq \{0\}$.
\item $M$ is called a \emph{torsion module} if $M=\Tor(M)$.  Prove that every finite abelian group is a torsion $\Z$-module.  Give an example of an infinite abelian group that is a torsion $\Z$-module.
\end{enumalph}
\end{problab}

\end{document}