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\title{Math 728\\Fall 2004\\Homework 6}
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\centerline{\it Due Wednesday October 20, 2004.}

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\noindent 1. Let $R$ be commutative with identity and $M$ and $R-$module.
\begin{itemize}
\item[a)] (5 pt) Show if $M$ is free, then $M$ is flat.
\item[b)] (5 pt) Show that if $M$ is projective, then $M$ is flat.
\end{itemize}

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\noindent 2. Let $S\subseteq\mathbb{Z}$ be a multiplicatively closed subset that does not contain 0. Consider the $\mathbb{Z}-$module $\mathbb{Z}_S=\{n/s\vert n\in\mathbb{Z}, s\in S\}$.
\begin{itemize}
\item[a)] (5 pt) Compute $\mathbb{Z}_S\otimes_{\mathbb{Z}}\mathbb{Z}_{p^a}$ where $p\in\mathbb{Z}$ is a nonzero prime.
\item[b)] (5 pt) Generalize your result from part a) by computing $\mathbb{Z}_S\otimes_{\mathbb{Z}}A$ where $A$ is any finite abelian group.
\item[c)] (5 pt) Generalize the results from a) and b) by computing $\mathbb{Z}_S\otimes_{\mathbb{Z}}G$ where $G$ is any finitely generated abelian group.
\end{itemize}

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\noindent 3. (5 pt) Show that $\mathbb{Z}_m\otimes_{\mathbb{Z}}\mathbb{Z}_n\cong\mathbb{Z}_{\text{gcd}(m,n)}$.

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\noindent 4. (5 pt) Show that if $P$ and $Q$ are projective $R-$modules, then $P\otimes_R Q$ is a projective $R-$module. Is the converse true?

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\noindent 5. (5 pt) Show that $\mathbb{Q}$ is a flat $\mathbb{Z}-$module which is not projective.
 



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