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\begin{document}
\begin{center}
    \textsc{Math 4140/5140. Homework 1}\\
    Due Wednesday, January 19
\end{center}

\vspace{1em}

\noindent \textbf{Note:} All numbered exercises are from Erdmann--Holm ([EH]).\\

\begin{enumerate}
    \item Read Sections 1.1 and 1.2 of $\eh$. 
    \item Let $G, H$ be groups and let $\varphi: G\ra H$ be a map such that
        $\varphi(gg')=\varphi(g)\varphi(g')$ for all $g,g'\in G$. Prove that $\varphi(1_G)=1_H$. 
    \item Let $R$ be a ring. Prove that $0\cdot r=0=r\cdot 0$ for all $r\in R$. 
    \item Prove that for a positive integer $n$, the set $M_n(\R)$ forms a commutative ring under the usual addition and matrix multiplication
        if and only if $n=1$. 
    \item Recall the statements of the first two isomorphism theorems for groups, then use the first isomorphism
        theorem to prove the second. 
    %\item  Exercise 1.4.
    %\item  Exercise 1.5.
    %\item  Exercise 1.8.
    %\item  Exercise 1.9.
\end{enumerate}
\end{document}


