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\title[101 Homework \#19]{Math 101: Algebra I \\ Homework, Day \#19}
\date{Assigned Wednesday, 12 October 2016; due Friday, 14 October 2016}

\maketitle

Let $R$ be a ring and let $M$ be a (left) $R$-module.  

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\maketitle

\begin{problab}{JV19.A} 
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 an integral 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}

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