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\title{Math 421-621\\Spring 2013\\Homework 4}
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\centerline{\it Due Wednesday, February 6, 2013.}


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\noindent 1. (5 pt) Let $F$ be a free $R-$module with basis $\{e_i\}_{i\in \Lambda}$, and $M$ another $R-$module. If $\{x_i\}_{i\in \Lambda}$ is a collection of elements of $M$, show that there is a unique homomorphism $\phi:F\longrightarrow M$ such that $\phi(e_i)=x_i$ for all $i\in\Lambda$.



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\noindent 2. We say that the $R-$module $P$ is {\it projective} if there is a free $R-$module $F$ and another $R-$module $K$ such that $P\oplus K\cong F$. 
\begin{itemize}
\item[a)] (5 pt) Show that if $F$ is free then it is projective.
\item[b)] (5 pt) Explain why the module $K$ is also projective.
\item[c)] (5 pt) Show that $\mathbb{Q}$ is not a projective $\mathbb{Z}-$module,
\item[d)] (5 pt) Give an example of a projective $R-$module that is not free.
\end{itemize}


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\noindent 3. Consider the following diagram of $R-$modules (the homomorphism $g$ is surjective).



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$\xymatrix{ & P\ar@{-->}[dl]_h\ar[d]^f
& \\
A\ar[r]_g &B\ar[r] & 0}$
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\begin{itemize}
\item[a)] (5 pt) Show that if $P$ is free, then there is an $R-$module homomorphism $h:P\longrightarrow A$ such that $gh=f$.
\item[b)] (5 pt) Show that if $P$ is projective, then there is an $R-$module homomorphism $h:P\longrightarrow A$ such that $gh=f$.
\end{itemize}


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\noindent 4. (5 pt) Show that if $P$ and $Q$ are projective $R-$modules, then $P\otimes_R Q$ is also projective.



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