\documentclass[12pt]{amsart}

\newcommand{\defi}[1]{\textsf{#1}} 

\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{\End}{End}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\img}{img}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\ev}{ev}

\usepackage{fullpage}

\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#13}
\date{Assigned Monday, 9 October 2017; due Wednesday, 11 October 2017}

\maketitle

\begin{problab}{13.1}
Let $R$ be a commutative ring.  Show that the map $R \to \End_R(M)$ where $r \in R$ maps to the multiplication-by-$r$ endomorphism
\begin{align*}
\phi_r : M &\to M \\
m &\mapsto rm
\end{align*}
is a ring homomorphism, and thereby that $\End_R(M)$ has the structure of an $R$-algebra.  (What happens if $R$ is not commutative?)
\end{problab}

\begin{problab}{13.2} 
Let $R$ be commutative ring.  
\begin{enumalph}
\item Prove that $\Hom_R(R,M) \simeq M$ as $R$-modules.  
\item Consider $R$ as an $R$-module.  Prove that there is a ring isomorphism $\End_R(R) \simeq R$.
\end{enumalph}
\end{problab}

\end{document}