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\hfill \textbf{March 2, 2012}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 3\hspace{1 cm}
Math 6303}}}

\vspace {1 cm}
\begin{enumerate}
\item
Show that the class $\mathcal{T}$ of all
abelian torsion groups admits products as
well as  sums.
\item Is $i_1:\mbb{Z}\rightarrow
\mbb{Z}\oplus\mbb{Z},n\mapsto (n,0),
i_2:\mbb{Z}\rightarrow \mbb{Z}\oplus
\mbb{Z},n\mapsto (0,n)$ a
 sum-system for the class of all groups?
Explain your answer!
\item Find $\mbb{Z}/(36)(3)$.
\item  Let $\mbf{A}=\mbb{Z}\oplus \mbb{Z}$ be
the free $\mbb{Z}-$module of rank 2. Let $\Delta=\{(n,n)\mid\in
\mbb{Z}\}$. Show that $\Delta$ is a submodule as well as a
homomorphic image of $\mbf{A}$. Is $\Delta$ adirect summand of
$\mbf{A}$?
\item If $\mbf{A}$ is
the free abelian group, freely generated by a countably infinite
set, then one has that $\mbf{A}\oplus\mbf{Z}\oplus
\mbf{Z}\iso\mbf{A}\oplus\mbb{Z}$ but $\mbb{Z}\oplus \mbb{Z}$ and
$\mbb{Z}$ are not isomorphic.
\end{enumerate} \end{document}
