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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 3.2  \\ Fall 2019 \hfill Due September 18}}}

\large

\vspace{1cm}

\begin{exercise}
Let $\varphi : R \to R'$ be a homomorphism of rings. 

\begin{itemize}

\item[\bf a.]  If $J$ is an ideal in $R'$, prove that $\varphi^{-1}(J)$ is an ideal in $R$.

\item[\bf b.]  If $I$ is an ideal in $R$ and $\varphi$ is surjective, prove that $\varphi(I)$ is an ideal in $R'$.

\end{itemize}
\end{exercise}

\begin{exercise}
A collection $\mathcal{C}$ of ideals in a ring $R$ is called a {\em chain} if for all $I, J \in \mathcal{C}$, either $I \subset J$ or $J \subset I$.  Let $\mathcal{C}$ be a chain of ideals in $R$.  

\begin{itemize}

\item[\bf a.]  Prove that $\displaystyle J = \bigcup_{I \in \mathcal{C}} I$ is an ideal in $R$.

\item[\bf b.]  If every ideal in $\mathcal{C}$ is proper, prove that $J$ is, too.

\end{itemize}
\end{exercise}

\begin{exercise}
Let $F$ be a field.  A surjective homomorphism $\nu : (F^{\times}, \cdot) \to (\Z,+)$ is called a {\em discrete valuation} if $\nu(a+b) \ge \min \{ \nu(a), \nu(b) \}$ for all $a, b \in F^{\times}$ (with $a \ne -b$).  Note that if we set $\nu(0) = \infty$, then the multiplicative and additive properties of $\nu$ extend to all of $F$.  

\begin{itemize}

\item[\bf a.]  Prove that $R = \{a \in F \, | \, \nu(a) \ge 0 \}$ is a subring of $F$, the {\em valuation ring}.

\item[\bf b.]  Show that $R^{\times} = \ker \nu$.

\item[\bf c.]  Show that for every $n \in \N$, $I_n = \{a \in F \, | \, \nu(a) \ge n \}$ is a proper ideal of $R$.  $I_1$ is called the {\em valuation ideal}.

\end{itemize}
\end{exercise}

\begin{exercise}
Let $F$ be a field with discrete valuation $\nu$, valuation ring $R$, and ideals $I_n$ as in the previous exercise.  A {\em uniformizer} of $R$ is an element $\pi \in F$ with $\nu(\pi) = 1$.  

\begin{itemize}

\item[\bf a.]  Let $\pi$ be a uniformizer of $R$.  Prove that $I_n = \pi^n R$.

\item[\bf b.]  Prove that $\{ 0 \} \subsetneqq \cdots \subsetneqq I_3 \subsetneqq I_2 \subsetneqq I_1 \subsetneqq R$.


\item[\bf c.]  Show that $\displaystyle \bigcap_{n \in \N} I_n = \{ 0 \}$.


\item[\bf d.]  Let $A \subset R$ be a nonzero proper ideal.  Show that there is an $n \in \N$ so that $A = I_n$.

\end{itemize}

\end{exercise}


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