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\title{Math 724\\Fall 2005\\Homework 6}
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\centerline{\it Due Wednesday December 7, 2005.}

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\noindent 1. (5 pt) Suppose that $R$ is a Dedekind domain with class group isomorphic to $\mathbb{Z}_4$. Additionally, assume that all the prime ideals of $R$ are in the principal class and the class of order 2. Show that $R$ is an HFD.

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\noindent 2. (5 pt) Let $R\subseteq T$ be an extension of rings. Show that any minimal prime ideal of $R$ is lain over by a prime ideal of $T$ (that is, if $\mathfrak{P}$ is a minimal prime ideal of $R$, then there is a prime ideal $\mathfrak{Q}\subseteq T$ such that $\mathfrak{Q}\bigcap R=\mathfrak{P}$).

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\noindent 3. (5 pt) Show that $R\subseteq R[x]$ is LO and GD, but is never INC. Show that this extension fails to be GU is $\text{dim}(R)\geq 1$.

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\noindent 4. Let $R$ be a domain and $I$ a fractional ideal of $R$.

\begin{itemize}
\item[a)] (5 pt) Show that $I$ is invertible if and only if $I$ is a projective $R-$module.
\item[b)] (5 pt) Show that if $I$ is invertible, it is of rank 1 (that is, there is an $R-$module, $J$, such that $I\otimes_R J\cong R$).
\item[c)] (5 pt) Show that if $I$ and $J$ are rank 1 projective $R-$modules, then $I\otimes_R J$ is also a rank 1 projective $R-$ module. Conclude that the set of isomorphism classes of  rank 1 projective $R-$modules forms a group with multiplication given by $\otimes_R$ (this group is called the Picard group of $R$ and is denoted by $\text{Pic}(R)$). 
\end{itemize}

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