\documentclass[12pt]{amsart}

\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amsxtra}
\usepackage{mathrsfs}
\usepackage[all]{xy}

\setlength{\hfuzz}{4pt}

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

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

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

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

\newcommand{\C}{\mathbb C}
\newcommand{\F}{\mathbb F}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}

\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Tor}{Tor}

\usepackage{fullpage}

\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#17}
\date{Assigned Wednesday, 18 October 2017; due Friday, 20 October 2017}

\maketitle

\vspace{1ex}

\begin{problab}{17.1} 
Let $R$ be a ring and let
\[ 
\xymatrix{
0 \ar[r] & M \ar^{\psi}[r] \ar[d]^{\alpha} & N \ar^{\phi}[r] \ar[d]^{\beta} & Q \ar[r] \ar[d]^{\gamma} & 0 \\
0 \ar[r] & M' \ar^{\psi'}[r] & N' \ar^{\phi'}[r] & Q' \ar[r] & 0 
} \]
be a homomorphism of short exact sequences.  Suppose that $\alpha$ and $\gamma$ are surjective.  Show that $\beta$ is surjective.
\end{problab}

\begin{problab}{17.2}
Let $R$ be a ring and let
\[ 
\xymatrix{
0 \ar[r] & M \ar^{\psi}[r] & N \ar^{\phi}[r] & Q \ar[r] & 0
} \]
be a short exact sequence.  Let $D$ be an $R$-module.  Show that the map
\[ \Hom_R(D,M) \to \Hom_R(D,N) \to \Hom_R(D,Q) \]
is exact.  \emph{[Hint: This is Theorem 28 on page 387 in Dummit and Foote.]}
\end{problab}

\end{document}