\documentclass[11pt]{article}
\usepackage{amsmath,amsfonts}

\title{Homework 6}
\author{Due Wednesday, November 9, 2016}
\date{}

\newcommand \C {\mathbb C}
\newcommand \Z {\mathbb Z}
\renewcommand \phi \varphi

\begin{document}
\maketitle

\begin{enumerate}

\item Let $R$ be a commutative ring and let $I, J \subset R$ be ideals.  Earlier you showed that $I+J$, $IJ$, and $I \cap J$ are ideals, and that $IJ \subset I \cap J$.
\begin{enumerate}
\item Show that if $I+J = (1)$ then $IJ = I \cap J$.
\item Show that the converse of part (a) does not hold in general by taking $R = \Z[x]$, $I = (2)$, and $J = (x)$.
\end{enumerate}

\item Let $R$ be a commutative ring, let $a, b, c, d \in R$, and consider the ideals $I = (a,b)$ and $J = (c,d)$.
\begin{enumerate}
\item Show that $I+J = (a, b, c, d)$.
\item Show that $IJ = (ac, ad, bc, bd)$.
\end{enumerate}

\item Let $R = \Z[\sqrt{-5}]$, and consider the ideals
\begin{align*}
I_1 &= (2,1 + \sqrt{-5}) \\
I_2 &= (2,1 - \sqrt{-5}) \\
I_3 &= (1 + \sqrt{-5},1 - \sqrt{-5}) \\[1ex]
J_1 &= (3, 1 + \sqrt{-5}) \\
J_2 &= (3, 1 - \sqrt{-5}).
\end{align*}
\begin{enumerate}
\item Show that $I_1 = I_2 = I_3$.
\item Show that $I_1 + J_1$, $I_1 + J_2$, and $J_1 + J_2$ are all equal to $(1)$.
\item Find $J_1 J_2$, $I_1 J_1$, $I_2 J_2$, and $I_1 I_2$.  Hint: They are all principal ideals, i.e.\ they can be generated by one element.
\end{enumerate}

\newpage

\item Let $I_1, J_1, J_2 \subset R$ be as in the previous problem.
\begin{enumerate}
\item Show that $R/I_1 \cong \Z/2$.  Hint: In lecture, in week 4 or so when we first discussed ideals, we wrote down a homomorphism $\phi\colon R \to \Z/2$ with $\ker \phi = I_1$.
\item Show that $R/J_1 \cong \Z/3$ and $R/J_2 \cong \Z/3$.
\item Show that $R/(3) \cong \Z/3 \times \Z/3$.

Hint: Use the Chinese Remainder Theorem.
\end{enumerate}

\item What is one question you have about last week's lectures?

\end{enumerate}

\end{document}
