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\title{Math 421-621\\Fall 2013\\Homework 3}
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\centerline{\it Due Wednesday, January 30, 2013.}


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\noindent 1. (5 pt) Let $A$ be an $R-$module and $\phi:A\longrightarrow A$ be an $R-$module homomorphism satisfying $\phi^2(a)=\phi(a)$ for all $a\in A$. Show that 

\[
A\cong \text{ker}(\phi)\oplus \text{im}(\phi).
\]


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\noindent 2. Let $M$ be an $R-$module with submodules $A$ and $B$.
\begin{itemize}
\item[a)] (5 pt) If $A\cong B$ show that $M/A\cong M/B$ or give a counterexample.
\item[b)] (5 pt) If there are $R-$module monomorphisms $\phi:A\longrightarrow B$ and $\psi:B\longrightarrow A$ show that $A\cong B$ or give a counterexample.
\item[c)] (5 pt) Under what conditions is $A\oplus B$ a submodule of $M$ and when is $A\oplus B=M$? 
\end{itemize}

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\noindent 3. Let $M$ be an $R-$module and $I\subset R$ a proper ideal. 
\begin{itemize}
\item[a)] (5 pt) Show that $IM=\{\sum_{i=1}^n\alpha_im_i\vert \alpha_i\in I,\ m_i\in M\}$ is an $R-$submodule of $M$.
\item[b)] (5 pt) Show that $M/IM$ is an $R/I-$module.
\end{itemize}

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\noindent 4. (5 pt) Show that a submodule of a free $R-$module need not be free. Give an example where $R$ is an integral domain and one where $R$ is a finite ring.


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