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\begin{document}
\begin{center}
    \textsc{Math 4140. Homework 11}\\
\end{center}

\vspace{1em}

There are no more homework problems, but please carefully read Chapters 1--6 of the textbook, our lecture notes,
and your previous homework problems again. Reflect on the material, and let me know if you have any questions.

The final exam will be available on Canvas from 11:59 a.m. of Saturday, May 1 to to 11:59 p.m. of Sunday, May 2.
The exam will be cumulative. It will contain six to eight problems that are similar in format to those on the
midterm. 

%\begin{enumerate}
    %\item Read Section 2.2-2.4. 
       %\item Assuming the isomorphism theorems for groups, prove the isomorphism theorems for modules, i.e., the
        %statements of Theorem 2.24. 
    %\item Let $R$ be a ring, let $M$ be an $R$-module, and let $(U_i)_{i\in I}$ be a family of $R$-submodules of $M$. Let
        %$N$ be the external direct sum of $(U_i)_{i\in I}$ defined in Part (b) of Definition 2.17, and
        %suppose that $M$ equals the internal direct sum of $(U_i)_{i\in I}$ in the sense that it satisfies the
        %conditions in Part (b) of Definition 2.15. Prove that the map $\phi: N\ra M$ defined by
        %$(u_i)_{i\in I}\mapsto \sum_{i\in I}u_i$ is an isomorphism of $R$-modules.
    %\item Exercise 2.5.
    %\item Exercise 2.6.
    %\item Exercise 2.15.
    %\item Exercise 2.22.
    %%\item Exercise 2.23.
        %\end{enumerate}
\end{document}


