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\title{Math 726\\Summer 2005\\Homework 2}
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\centerline{\it Due Tuesday July 5, 2005.}

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\noindent 1. 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 2. (5 pt) Show that it is not true in general that $\prod_{\alpha\in I}(A_{\alpha}\otimes_R B)$ is isomorphic to $(\prod_{\alpha\in I} A_{\alpha})\otimes_R B$.

\centerline{}

\noindent 3. (5 pt) Let $F$ be an additive functor from the category of $R-$modules to itself. Show that if the sequence

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$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$
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\noindent is split exact then $F(A)$ is a summand of $F(B)$. Use this to show that $F$ preserves finite sums.

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\noindent 4. (5 pt) Show that $\text{Hom}_R(P,-)$ is an exact functor (from the category of $R-$modules to itself) if and only if $P$ is a projective $R-$module.






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