\documentclass[12pt,letterpaper]{hmcpset}
\usepackage[margin=1in]{geometry}
\pagestyle{empty}

\name{}
\class{Math 171 - Abstract Algebra I Section \_\!\_\!\_\!\_}
\assignment{HW \#9}
\duedate{10/7/10}

\newcommand{\Z}{\mathbb{Z}}

\begin{document}

\problemlist{Dummit \& Foote (7.1) 14, 15, 16, 17}

\begin{problem}[7.1.14]
Let $x$ be a nilpotent element of the commutative ring $R$ (cf. the preceding exercise).
\begin{itemize}
\item[(a)] Prove that $x$ is either zero or a zero divisor.
\item[(b)] Prove that $rx$ is nilpotent for all $r \in R$.
\item[(c)] Prove that $1+x$ is a unit in $R$.
\item[(d)] Deduce that the sum of a nilpotent element and a unit is a unit.
\end{itemize}
\end{problem}
\newpage

\begin{problem}[7.1.15]
A ring $R$ is called a \emph{Boolean ring} if $a^2=a$ for all $a \in R$.  Prove that every Boolean ring is commutative.
\end{problem}
\newpage

\begin{problem}[7.1.16]
Prove that a Boolean ring that is an integral domain has only two elements.
\end{problem}
\newpage

\begin{problem}[7.1.17]
Let $R$ and $S$ be rings.  Prove that the direct product $R \times S$ is a ring under componentwise addition and multiplication.  Prove that $R \times S$ is commutative if and only if both $R$ and $S$ are commutative.  Prove that $R \times S$ has an identity if and only if both $R$ and $S$ have identities.
\end{problem}

\end{document}