\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{\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 \#5}
\date{Assigned Wednesday, 25 October 2017; due Wednesday, 1 November 2017}

\maketitle

\vspace{1ex}

\begin{problab}{W5.1}
Let $R$ be a ring and let  
\begin{equation*}
0 \to M \xrightarrow{\psi} N \xrightarrow{\phi} Q \to 0  \tag{$*$}
\end{equation*}
 be a short exact sequence of (left) $R$-modules.  A \emph{retraction} of $(*)$ is an $R$-module homomorphism $\rho\colon N \to M$ such that $\rho \circ \psi = \id_M$.  

Show that $(*)$ is split if and only if $(*)$ has a retraction. 
\end{problab}

\begin{problab}{W5.2}
Let $k$ be a field, let $R = \begin{pmatrix} k & k \\ 0 & k \end{pmatrix} \subseteq \M_2(k)$ be the subring of upper-triangular matrices.  Let $P=\begin{pmatrix} k & 0 \\ 0 & 0 \end{pmatrix}$ and $Q=\begin{pmatrix} 0 & k \\ 0 & k \end{pmatrix}$.  Show that $P$ and $Q$ are projective left $R$-modules that are not free. 
\end{problab}

\begin{problab}{W5.3}
Let $R$ be a ring and let $P$ be a projective (left) $R$-module that has $R$ as a direct summand.  Show that if $P \oplus R^m \simeq R^n$ with $n > m$, then $P^{m+1}$ is free.  
\end{problab}

\begin{problab}{W5.4}
Let $f\colon \Z^n \to \Z^n$ be a homomorphism of groups.

\begin{enumalph}
\item Show that if $f$ is surjective, then $f$ is injective.
\item Show that if $f$ is injective, then $\coker f$ is a finite group.
\end{enumalph}
\end{problab}

\end{document}