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\title{Math 724\\Summer 2010\\Homework 2}
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\centerline{\it Due Monday, July 17, 2010.}

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\noindent 1. (5 pt) Show that the following are equivalent.
\begin{enumerate}
\item INC holds.
\item If $\mathfrak{P}\subseteq R$ is a prime ideal and $\mathfrak{Q}\subseteq T$ contracting to $\mathfrak{P}$ then $\mathfrak{Q}$ is maximal with respect to the exclusion of $S$, the complement of $\mathfrak{P}$ in $R$.
\end{enumerate}

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\noindent 2. (5 pt) Give an example of an extension that is LO but not GU (or prove that GU and LO are equivalent).

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\noindent 3. (5 pt) Prove that $R$ is Dedekind if and only if every nonzero proper ideal can be written as the product of prime ideals.




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