\documentclass[12pt]{amsart}

% \include{d:/TeX/hwhead}
% \include{d:/TeX/hwops}

\usepackage{fullpage}
\usepackage[all]{xy}

\newenvironment{problab}[1]
{\noindent\textbf{Problem #1}.}
{\vskip 6pt}

\newenvironment{enumalph}
{\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\alph{enumi})}}}
{\end{enumerate}}

\newenvironment{enumroman}
{\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}}
{\end{enumerate}}

\theoremstyle{remark}
\newtheorem*{solu}{Solution}

\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amsxtra}

\setlength{\hfuzz}{4pt}

\newcommand{\calC}{\mathcal{C}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\bfA}{\mathbb A}
\newcommand{\bfC}{\mathbb C}
\newcommand{\C}{\mathbb C}
\newcommand{\Q}{\mathbb Q}
\newcommand{\bfQ}{\mathbb Q}
\newcommand{\bfR}{\mathbb R}
\newcommand{\R}{\mathbb R}
\newcommand{\bfZ}{\mathbb Z}
\newcommand{\Z}{\mathbb Z}

\DeclareMathOperator{\M}{M}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\opchar}{char}
\DeclareMathOperator{\opspan}{span}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\repart}{Re}
\DeclareMathOperator{\impart}{Im}
\DeclareMathOperator{\img}{img}

\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Weekly Homework \#6}
\date{Assigned Friday, 3 November 2017; due Friday, 10 November 2017}

\maketitle

\vspace{1ex}

\begin{problab}{W6.1}
Let $R$ be a commutative ring, let $S \subset R$ be a multiplicatively closed set containing $1$, and let $S^{-1} R = R[S^{-1}]$ be the localization at $S$.  Let $\phi\colon R \to S^{-1} R$ be the ring homomorphism $r \mapsto r/1$.  
\begin{enumalph}
\item Let $I \subseteq R$ be an ideal.  Then $I$ is an $R$-module, so we have defined $S^{-1} I \subseteq S^{-1} R$, and
\[ S^{-1} I = \{a/s : a \in I, s \in S\}. \]
Show that $S^{-1} I$ is an ideal in $S^{-1} R$.  Show that $S^{-1} I = S^{-1} R$ if and only if $I \cap S \neq \emptyset$. 
\item Show that every ideal $I' \subseteq S^{-1} R$ is of the form $I' = S^{-1} I$ for an ideal $I \subseteq R$.
\item Show that there is a bijection between the \emph{prime} ideals of $S^{-1} R$ and the prime ideals of $R$ disjoint from $S$.
\end{enumalph}
\end{problab}

\begin{problab}{W6.2}
\begin{enumalph}
\item Let $R$ be a Euclidean domain with norm $N$.  Let 
\[ m=\min(\{ N(a) : a \in R, a \neq 0\}). \]
Show that every nonzero $a \in R$ with $N(a)=m$ is a unit in $R$.  Deduce that a nonzero element of norm zero in $R$ is a unit; show by an example that the converse of this statement is false.
\item Let $F$ be a field and let $R=F[[x]]$.  Show that $R$ is Euclidean.  What does part (a) tell you about $R^\times$?  What are the irreducibles in $R$, up to associates?
\end{enumalph}
\end{problab}

\begin{problab}{W6.3}
Let $R$ be a domain and let $M$ be an $R$-module.  Elements $x_1,\dots,x_n \in M$ are \emph{$R$-linearly independent} if whenever $a_1x_1+\dots+a_nx_n=0$ with $a_i \in R$, then $a_1=\dots=a_n=0$.  

The \emph{rank} of $M$ is the maximal number of $R$-linearly independent elements of $M$.  

\begin{enumalph}
\item Suppose that $M$ has rank $n$ and that $x_1,\dots,x_n$ is any maximal set of $R$-linearly independent elements of $M$.  Let $N=Rx_1+\dots+Rx_n$ be the $R$-submodule generated by $x_1,\dots,x_n$.  Prove that $N$ is isomorphic to $R^n$ and that the quotient $M/N$ is a torsion $R$-module.  \emph{[Hint: Show that the map $R^n \to N$ which sends the $i$th standard basis vector to $x_i$ is an isomorphism of $R$-modules.]}
\item Prove conversely that if $M$ contains a submodule $N$ that is free of rank $n$ (i.e., $N \cong R^n$) such that the quotient $M/N$ is a torsion $R$-module then $M$ has rank $n$.  \emph{[Hint: Let $y_1,\dots,y_{n+1}$ be any $n+1$ elements of $M$.  Use the fact that $M/N$ is torsion to write $r_iy_i$ as a linear combination of a basis for $N$ for some nonzero elements $r_i$ of $R$.  Use an argument like Proposition\/ $12.1.3$ to show that the $r_iy_i$, and hence also the $y_i$, are linearly dependent.]}
\item Let $R=\Z[x]$ and let $M=(2,x)$ be the ideal generated by $2$ and $x$, considered as a submodule of $R$.  Show that $\{2,x\}$ is not a basis of $M$.  Show that the rank of $M$ is $1$ but that $M$ is not free of rank $1$.
\end{enumalph}
\end{problab}

\begin{problab}{W6.4}
\begin{enumalph}
\item Let $N \leq \Z^2$ be the submodule generated by $(2,4)$ and $(8,10)$.  Write $\Z^2/N$ as a product of cyclic groups.
\item Let $R$ be a PID.  Let $M \subseteq R^n$ be an $R$-submodule such that 
\[ \#(R^n/M)=[R^n:M]=p \] 
where $p \in \Z$ is prime and $p$ is a nonzerodivisor in $R$.  Show that $M$ is free of rank $n$ and there is a basis $x_1,\dots,x_n$ of $R^n$ and $q \in R$ such that $M=Rx_1 \oplus \dots \oplus Rq x_n$ and $[R:(q)]=p$.
\end{enumalph}
\end{problab}

\begin{problab}{W6.5}
\begin{enumalph}
\item Prove that two $2 \times 2$ matrices over $F$ which are not scalar matrices are similar if and only if they have the same characteristic polynomial.
\item Prove that two $3 \times 3$ matrices are similar if and only if they have the same characteristic and minimal polynomials.  Give an explicit counterexample to this assertion for $4 \times 4$ matrices.
\end{enumalph}
\end{problab}

\begin{problab}{W6.6}
Find all similarity classes of $6 \times 6$ matrices over $\Q$ with minimal polynomial $(x+2)^2(x-1)$.  [It suffices to give all lists of invariant factors and write out some of their corresponding matrices.]
\end{problab}

\end{document}