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\begin{center} {\large \textbf{Math 721 -- Homework 3}} \medskip \\ Due Friday, January 31, 2019 at 5pm  \\ \ \\ 
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\noindent Good practice problems (do not turn in solutions): \\
DF 10.5 Exercises  1, 2, 16, 22, 26

\bigskip \bigskip 



\begin{prob}[DF Exercise 3, 4, 5] Let $X_1$, $X_2$ be $R$-modules.  \smallskip 
\begin{itemize}
\item[(a)] Show that $X_1 \oplus X_2$ is projective if and only if $X_1$ and $X_2$ are projective.  \smallskip 
\item[(b)] Show that $X_1 \oplus X_2$ is injective if and only if $X_1$ and $X_2$ are injective.  \smallskip 
\item[(c)] Show that $X_1 \oplus X_2$ is flat if and only if $X_1$ and $X_2$ are flat.  
\end{itemize}
\end{prob}


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\begin{prob}[DF Exercise 14]  
Let $0\to A\xrightarrow{\varphi} B \xrightarrow{\psi} C \to 0$ be a sequence of $R$ modules.   \smallskip 
\begin{itemize}
\item[(a)] Show that  \[0\to \Hom_R(X,A)\xrightarrow{\varphi'} \Hom_R(X,B) \xrightarrow{\psi'} \Hom_R(X,C) \to 0\] is a 
short exact sequence of abelian groups for all $R$-modules $X$ if and only if the original sequence is a split short exact sequence. (Hint: consider $X = A$.) \\ 
\item[(b)] Show that  \[0\to \Hom_R(C,X) \xrightarrow{\psi'} \Hom_R(B,X) \xrightarrow{\varphi'} \Hom_R(A,X) \to 0\] is a 
short exact sequence of abelian groups for all $R$-modules $X$ if and only if the original sequence is a split short exact sequence.
\end{itemize}
\end{prob} 





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