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\title{Math 724\\Summer 2009\\Homework 1}
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\centerline{\it Due Sometime.}

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\noindent 1. (5 pt) Show that if $T_1:\mathfrak{A}\longrightarrow\mathfrak{B}$ and $T_2:\mathfrak{B}\longrightarrow\mathfrak{C}$ are both covariant or both contravariant (functors), then $T_2\circ T_1$ is covariant.

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\noindent 2. Give examples for each of the following.
\begin{itemize}
\item[a)] (5 pt) Show that $\text{Hom}_R(\prod_{\alpha\in I}A_{\alpha},B)$ is not generally isomorphic to $\prod_{\alpha\in I}\text{Hom}_R(A_{\alpha},B)$.
\item[b)] (5 pt) Show that $\text{Hom}_R(\prod_{\alpha\in I}A_{\alpha},B)$ is not generally isomorphic to $\oplus_{\alpha\in I}\text{Hom}_R(A_{\alpha},B)$.
\item[a)] (5 pt) Show that $\text{Hom}_R(B,\oplus_{\alpha\in I}A_{\alpha})$ is not generally isomorphic to $\oplus_{\alpha\in I}\text{Hom}_R(B, A_{\alpha})$.
\item[a)] (5 pt) Show that $\text{Hom}_R(B,\oplus_{\alpha\in I}A_{\alpha})$ is not generally isomorphic to $\prod_{\alpha\in I}\text{Hom}_R(B,A_{\alpha})$.
\end{itemize}

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\noindent 3. (20 pt) Given the diagram of $R-$modules below with exact bottom row, 
\begin{center}
$\xymatrix{ & P\ar@{-->}[dl]_h\ar[d]^f
& \\
A\ar[r]_g &B\ar[r] & 0}$
\end{center}
\noindent we say that $P$ is projective if there is an $R-$module homomorphism $h:P\longrightarrow A$ such that $gh=f$.

Prove the following conditions are equivalent:
\begin{itemize}
\item[a)] $P$ is projective.
\item[b)] Every short exact sequence of the form $\xymatrix@1{0\ar[r]&A\ar[r]&B\ar[r]&P\ar[r]&0}$ is split exact.
\item[c)] There is an $R-$module $K$ and a free module $F$ such that $F\cong P\oplus K$.
\item[d)] $\text{Hom}_R(P,-)$ is an exact functor.
\end{itemize}

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4. (20 pt) Dualize the previous problem to ``invent" injective modules and prove the equivalence of the analogous conditions.


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