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\title{Math 724\\Fall 2005\\Homework 1}
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\centerline{\it Due Monday September 5, 2005.}

\centerline{}

\noindent 1. Let $R$ be a commutative ring with identity and $\mathfrak{N}\subseteq R$ the set of nilpotent elements of $R$.
\begin{itemize}
\item[a)] (5 pt) Show that $\mathfrak{N}$ is an ideal.
\item[b)] (5 pt) Show that \[
\mathfrak{N}=\bigcap_{\mathfrak{P}:\text{prime}}\mathfrak{P}.
\]
\item[c)] (5 pt) Generalize part b) and show that if $I\subseteq R$ is an ideal, then 
\[
\sqrt{I}=\bigcap_{\mathfrak{P}\supseteq I}\mathfrak{P},
\]
\noindent where the intersection is taken over all prime ideals containing $I$.
\end{itemize}

\centerline{}

\noindent 2. (5 pt) Let $R$ be a commutative ring with identity and $I,J\subseteq R$ ideals. We define $(I:J)=\{r\in R\vert rJ\subseteq I\}$. Show that $(I:J)$ is an ideal of $R$.

\centerline{}

\noindent 3. Let $R$ be a commutative ring with identity.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ contains a non-principal ideal, then $R$ contains an ideal, $I$, that is maximal with respect to being non-principal.
\item[b)] (5 pt) Show that any ideal that is maximal with respect to being non-principal is prime.
\item[c)] (5 pt) Show that if $R$ contains an ideal that is not finitely generated, then $R$ contains an ideal, $J$, that is maximal with respect to being not finitely generated.
\item[d)] (5 pt) Show that any ideal that is maximal with respect to being not finitely generated is prime.
\item[e)] (5 pt) Show that $R$ is a PID if and only if every prime ideal of $R$ is principal.
\item[f)] (5 pt) Show that $R$ is Noetherian (every ideal is finitely generated) if and only if every prime ideal is finitely generated.
\end{itemize} 

\centerline{}

\noindent 4. (5 pt) Let $R$ be a integral domain. Show that the following conditions are equivalent.
\begin{enumerate}
\item Every $R-$module is projective.
\item Every $R-$module is injective.
\item Every $R-$module is free.
\item $R$ is a field.
\end{enumerate}
{\it Note that 1)$\Longleftrightarrow$ 2) and does not require the assumption integral domain, can you give an example of a commutative ring with 1 such that every ideal is projective (and injective) but not every module is free?}

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