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\title{Math 720\\Fall 2010\\Homework 8}
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\centerline{\it Due Friday, December 10, 2010.}

\centerline{}

\noindent 1. Let $d$ be a square free integer. We define a quadratic ring of integers to be $R:=\mathbb{Z}[\omega]=\{a+b\omega\vert a,b\in\mathbb{Z}\}$ where
\[
\omega=
\begin{cases}
\sqrt{d} &\text{ if $d\equiv 2,3\text{mod}(4)$}\\
\frac{1+\sqrt{d}}{2} &\text{ if $d\equiv 1\text{mod}(4)$}.
\end{cases}
\]

We define the norm map $N:R\longrightarrow\mathbb{Z}$ by $N(a+b\omega)=(a+b\omega)(a+b\overline{\omega})$ where 

\[
\overline{\omega}=
\begin{cases}
-\sqrt{d} &\text{ if $d\equiv 2,3\text{mod}(4)$}\\
\frac{1-\sqrt{d}}{2} &\text{ if $d\equiv 1\text{mod}(4)$}.
\end{cases}
\]

\noindent Prove the following properties of the norm.
\begin{itemize}
\item[a)] (5 pt) $N(xy)=N(x)N(y)$ for all $x,y\in R$.
\item[b)] (5 pt) $N(x)=0$ if and only if $x=0$.
\item[c)] (5 pt) $x\in U(R)$ if and only if $N(x)=\pm 1$.
\item[d)] (5 pt) Use the norm map to show that $\mathbb{Z}[\sqrt{-14}]$ is not an HFD.
\item[e)] (5 pt) Use the norm map to show that the ring $\mathbb{Z}[\sqrt{10}]$ is not a UFD.
\end{itemize}

\centerline{}


\noindent 2. (5 pt) Let $R$ be commutative with identity. We say that $R$ is Von Neumann regular if for all $a\in R$ there is an $x\in R$ such that $a^2x=a$. Let $R$ be Von Neumann regular and let $\mathfrak{P}$ be a prime ideal of $R$. Show that $R/\mathfrak{P}\cong R_{\mathfrak{P}}$.

\centerline{}

\noindent 3. (5 pt) Show that any overring of the integers $\mathbb{Z}$ is of the form 
$\mathbb{Z}_S$ for some multiplicatively closed subset of 
$\mathbb{Z}$. (Note: an overring of a domain $D$, with quotient field $K$,
is a ring $R$ such that $D\subseteq R\subseteq K$.)

\centerline{}

\noindent 4. (5 pt) Let $R$ be commutative with identity. Show that the following conditions are equivalent.
\begin{itemize}
\item[a)] $R$ has a unique prime ideal.
\item[b)] Every nonunit of $R$ is nilpotent.
\item[c)] $R$ has a minimal prime ideal which contains all zero divisors and all nonunits of $R$ are zero-divisors.
\end{itemize}

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