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\begin{document}
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\begin{center}
{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 7 \small{(due Thursday, Nov 19)}
\end{center}

\begin{enumerate}
\item Let $A = \Z$ and $\p = p\Z$ with $p$ a prime in $\Z$.  We have
  characterized the localization $A_\p = \Z_\p$ as $\{a/b \in \Q \mid
  a,b \in \Z,\ p\nmid b,\ \gcd(a,b) = 1\}$.
  \begin{enumerate}

  \item Show that every nonzero element in $\Z_\p$ can be written
    uniquely as $p^\nu u$ where $\nu$ is a nonnegative integer and $u
    \in \Z_\p^\times$.  You may of course assume unique factorization
    in $\Z$.

  \item Characterize all the ideals of $\Z_\p$, and confirm that
    $\Z_\p$ has a unique maximal ideal.

  \item Show that $\Z_\p/ p\Z_\p \cong \Z/p\Z$.

  \end{enumerate}


\item Let $A$ be a commutative ring with identity.
  \begin{enumerate}
  \item Suppose that for each prime ideal $\P$ in $A$, the local ring
    $A_\P$ has no nonzero nilpotent elements. Show that $A$ has no
    nonzero nilpotent elements.  \textit{Hint:} Show that for an
    element $x \in A$, the set $\Ann(x) = \{ y \in A \mid yx = 0\}$ is
    an ideal of $A$. $\Ann(x)$ is called the annihilator of the
    element $x$.

  \item Proof or counterexample: If for each prime $\P$ of $A$, each
    localization $A_\P$ is an integral domain, then $A$ is an integral
    domain.
  \end{enumerate}

\item Let $A$ be an integral domain, $S \subsetneq A$ a multiplicative
  subset containing 1 (but not containing 0).
  \begin{enumerate}
  \item Show that $S^{-1}A$ is an integral domain.
  \item Show that if $A$ is a PID (every ideal is principal), so is
    $S^{-1}A$.
  \end{enumerate}

\item Consider the localization of $\Z[x]$ at the prime ideal $(x)$.
  \begin{enumerate}
  \item Describe the elements of $\Z[x]_{(x)}$.


  \item Is $(x)$ maximal in $\Z[x]_{(x)}$?  If so, describe the
    resulting quotient field.



  \item How does $\Z[x]_{(x)}$ compare to $\Q[x]_{(x)}$?


  \end{enumerate}

\item Let $A$ be a commutative ring with identity, and let $X$ be the
  set of all prime ideals in $A$. $X$ is called the prime spectrum of
  $A$, written $Spec(A)$.  For each subset $E\subseteq A$, let $V(E)$
  denote the set of primes ideals of $A$ which contain $E$. The
  properties below demonstrate that the sets $V(E)$ satisfy the axioms
  for closed sets in a topological space.  This topology is called the
  Zariski topology on $Spec(A)$.

  Prove that:
  \begin{enumerate}
  \item If $I = \la E\ra$ is the ideal generated by $E$, then $V(I) =
    V(E)$.



  \item Show that $V(0) = X$ and $V(1) = \emptyset$.

  

  \item If $\{E_i\}_{i\in I}$ is any family of subsets of $A$, then
    $V(\cup_i E_i) = \cap_{i\in I}V(E_i)$.

 

  \item For any ideals $I,J$ of $A$, show that $V(I\cap J) = V(IJ) =
    V(I) \cup V(J)$.
 


  
  \end{enumerate}


\end{enumerate}







\end{document}