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\title{Math 721\\Spring 2011\\Homework 3}
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\centerline{\it Due Wednesday February 16, 2010.}

\centerline{}



\noindent 1. (15 pt) Show that the following conditions on an $R-$module $I$ are equivalent.
\begin{itemize}
\item[a)] $I$ is injective.
\item[b)] Every short exact sequence of the form
\begin{center}
$\xymatrix@1{0\ar[r] & I\ar[r] &B\ar[r] & C\ar[r] & 0}$
\end{center}
\noindent is split exact.
\item[c)] $I$ is a direct summand of any module of which it is a submodule.
\end{itemize} 

\centerline{}

\noindent 2. (10 pt) Show that the sequence of $R-$module homomorphisms

\begin{center}
$\xymatrix@1{A\ar[r]^f &B\ar[r]^g & C\ar[r] & 0}$
\end{center}

is exact if and only if for every $R-$module $D$, the sequence of $R-$module homomorphisms

\begin{center}
$\xymatrix@1{0\ar[r] &\text{Hom}_R(C,D)\ar[r]^{\overline{g}} &\text{Hom}_R(B,D)\ar[r]^{\overline{f}} &\text{Hom}_R(A,D)}$
\end{center}

\centerline{}


\noindent 3. Let $A$ be an abelian group, show that we have the following isomorphisms of abelian groups.
\begin{itemize}
\item[a)] (5 pt) $\text{Hom}_{\mathbb{Z}}(\mathbb{Z}_m, A)\cong A[m]$ where $A[m]=\{a\in A\vert ma=0\}$.
\item[b)] (5 pt) $\text{Hom}_{\mathbb{Z}}(\mathbb{Z}_m, \mathbb{Z}_n)\cong \mathbb{Z}_d$, where $d=\text{gcd}(m,n)$.
\end{itemize}

\centerline{}

\noindent 4. (5 pt) Let $\{J_i\}_{i\in I}$ be a family of $R-$modules. Show that $\prod_{i\in I}J_i$ is injective if and only if $J_i$ is injective for all $i\in I$.





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