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\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#9} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Nov. $20^\mathrm{th}$, 2020}

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\begin{description}
\item[\#1 Units and Zero Divisors] Let $R$ be a ring with $1$. Recall that $R^\times = \{ u \in R \;|\; u^{-1} \mbox{ exists}\}$ is the group of units. Also, if $a,b \in R$ such that $a\not=0$, $b\not=0$ but $ab=0$, then we call $a$ a {\it left zero divisor} and $b$ a {\it right zero divisor}. A {\it zero divisor} is either a left or right zero divisor (or both).

\begin{enumerate}[(a)]
\item Let $u \in R^\times$. Show that $-u \in R^\times$.
\item Suppose that $r \in R$ is a zero divisor. Show that $r$ is not a unit. 

{\it Note:} You need to consider two cases. One case where $r$ is a left zero divisor and another where $r$ is a right zero divisor.
\end{enumerate}

\item[\#2 Sticking to my Principals] Let $R$ be a commutative ring with $1$ and let $a \in R$. Recall that $(a) = \{ ra \;|\; r \in R\}$ is the principal ideal generated by $a$.

\begin{enumerate}[(a)]
\item List all of the (distinct) principal ideals in $\mathbb{Z}_{12}$. Show their contents. Example: $(0) = \{0\}$.
\item Let $R$ be an integral domain. Show that $(a)=(b)$ if and only if $a$ and $b$ are associates (i.e., $a=ub$ for some unit $u \in R^\times$).
\end{enumerate}

\item[\#3 Ideal Arithmetic] Let $R$ be ring and $I,J \triangleleft R$. One defines $I+J = \{x+y \;|\; x \in I \text{ and } y \in J\}$ to be the {\it sum} of $I$ and $J$. Similarly, one defines $IJ = \{ x_1y_1+\cdots+x_\ell y_\ell \;|\; \text{for some } \ell \geq 0 \text{ and } x_1,\dots,x_\ell \in I \text{ and } y_1,\dots,y_\ell \in J\}$ to be the {\it product} of $I$ and $J$. It can be shown that $I+J$,\  $IJ$, and $I \cap J$ are ideals of $R$.

\begin{enumerate}[(a)]
\item Prove that $I+J \triangleleft R$.
\item Consider $I=(12)$ and $J=(15)$ in $\mathbb{Z}$. Calculate $I+J$, $IJ$, and $I \cap J$. 

Since $\mathbb{Z}$ is a principal ideal domain, $(a)(b)=(c)$, $(a)+(b)=(d)$, and $(a) \cap (b)=(\ell)$ for some $c,d,\ell \in \mathbb{Z}$. Make a conjecture about how $a$ and $b$ are related to $c$, $d$, and $\ell$. 

{\it Note:} You don't have to prove your conjectures. Just tell me how you think this ``ideal arithmetic'' works.
\end{enumerate}

\item[\#4 Prime and Maximal] As a quick reminder, in $\mathbb{Z}$ and in $\mathbb{Z}_n$, we know that subgroup = normal subgroup = cyclic subgroup = subring = ideal = principal ideal.
\begin{enumerate}[(a)]
\item Find all the ideals of $\mathbb{Z}_{24}$ and draw the corresponding lattice. Which ideals are prime? Which are maximal?
\item Determine which ideals in $\mathbb{Z}$ are prime and which are maximal. [Prove your assertions.]

{\it Note:} Don't forget to consider the trivial ideal: $\{0\}$.
\end{enumerate}

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