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\title{Math 720\\Fall 2010\\Final Exam}
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{\it Due Wednesday, December 15, 2010.}
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\noindent 1. For this problem, $R$ will be a UFD with quotient field $K$.
\begin{itemize}
\item[a)] (5 pt) Show that $K[x]$ is a PID.
\item[b)] (5 pt) Show that if $\pi$ is a prime element of $R$, then $\pi$ is also a prime element of $R[x]$.
\item[c)] (5 pt) Show that if $\mathfrak{P}\subseteq R[x]$ is a prime ideal then $\mathfrak{P}\bigcap R$ is a prime ideal of $R$.
\item[d)] (5 pt) Show that $R[x]$ is a UFD (hint: show every nonzero prime ideal of $R[x]$ contains a nonzero prime element by considering the cases when $\mathfrak{P}\bigcap R$ is zero and nonzero). 
\end{itemize}


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\noindent 3. Let $R$ be commutative with identity.
\begin{itemize}
\item[a)] (5 pt) Show that if there is an ideal $I\subseteq R$ that is not finitely generated, then there is an ideal $J\subseteq R$ that is maximal with respect to the property of not being finitely generated.
\item[b)] (5 pt) Show that any ideal that is maximal with respect to the property of not being finitely generated is prime.
\item[c)] (5 pt) Use this to show that a ring is Noetherian if and only if every {\it prime} ideal is finitely generated.
\end{itemize}

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\noindent 4. (5 pt) Suppose that $R$ is commutative with identity. Show that if $R$ is Noetherian, then $R[x]$ is Noetherian. 


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