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\title{Math 724\\Fall 2005\\Homework 5}
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\centerline{\it Due Wednesday November 30, 2005.}

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\noindent 1. (5 pt) Show that if $R$ is a domain in which every nonzero proper ideal is a product of a finite number of prime ideals, then $R$ is Dedekind.

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\noindent 2. (5pt) Show that if $R$ is Dedekind, then every ideal can be generated with 2 elements.

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\noindent 3. (5 pt) Show that $R$ is a Pr\"{u}fer domain if and only if every two-generated ideal is invertible.

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\noindent 4. (5 pt) Show that if $R$ is a Pr\"{u}fer domain, $\mathfrak{P}\subseteq R$ is a prime ideal and $S\subseteq R$ is a multiplicatively closed set (not containing 0) then $R_S$ and $R/\mathfrak{P}$ and Pr\"{u}fer domains.

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\noindent 5. (5 pt) Show that if $R$ is a Pr\"{u}fer domain, then $R$ is a PID if and only if $R$ is a UFD.


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