\documentclass[11pt]{article}
\usepackage{amsfonts, amssymb}
\usepackage{latexsym}
\addtolength{\textwidth}{3.5cm}
\addtolength{\oddsidemargin}{-2cm}
\addtolength{\topmargin}{-3.cm}
\addtolength{\textheight}{2.25cm}
\newcommand{\mbf}[1]{\mathbf{#1}}
\newcommand{\mcal}[1]{\mathcal{#1}}
\pagestyle{empty}
\begin{document}

\hfill \textbf{October 29, 2014}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 5\hspace{1 cm}
Math 6302}}}

\vspace{2 cm}

 Do the following problems in
the book:
\begin{enumerate}
\item
p. 120, Exercises 3, 11, 14,
\item
p. 133, Exercises  2, 3, 13, 18
\item Find a non-commutative ring $A$ and elements $a,b \in A $ such that $a\cdot b=0 \mbox{ but }b\cdot a \ne 0$

\item
Assume that $\varphi,\psi:\mbf{A}\to \mbf{B}$ are two
homomorphisms that agree on a generating set $C$ of $A$.
Prove that $\varphi=\psi$.
 \item Let
$\mbf{D}=\{a/b\mid a,b\in \mathbb{Z},b\: \mbox{ odd}\}.$
Show: $\mbf{D}$ is a commutative domain. What are the
invertible elements of $\mbf{D}$? Prove that $\mbf{D}$ has a
unique maximal ideal.
\medskip

\item Let $C[0,1]$ be the ring of
continuous functions on the closed interval $[0,1]$ and let
$x \in [0,1].$ Prove that $M_x=\{f\mid f(x)=0\}$ is a
maximal ideal of $C[0,1].$ What can you say about the field
$C[0,1]/M_x?$ Is every maximal ideal of $C[0,1]$ of the form
$M_x$ for some $x?$


 \end{enumerate} \end{document}




 \end{enumerate}
\end{document}
