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\noindent
\parbox{2.25in}{\bf Math 4720 \& 5210}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Mar. $18^\mathrm{th}$, 2019}

\vspace{0.05in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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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.

\item Now suppose that $R$ is commutative. Let $\mathcal{L}_r:R\to R$ where $0 \not=r \in R$ and $\mathcal{L}_r(x)=rx$. So $\mathcal{L}_r$ is left (and also because $R$ is commutative, right) multiplication by $r$. Show that $\mathcal{L}_r$ is one-to-one and onto if and only if $r$ is a unit. Show that $\mathcal{L}_r$ is one-to-one if and only if $r$ is not a zero divisor.
\end{enumerate}

\item[\#2 A Centrist Approach] Let $R$ be a ring and define $Z(R) = \{ r \in R \;|\; rx=xr \mbox{ for all } x \in R\}$. This is the center of the ring $R$.
\begin{enumerate}[(a)]
\item Show that $Z(R)$ is a subring of $R$. Moreover, if $R$ is a ring with $1$, so is $Z(R)$.
\item Let $D$ be a division ring (i.e., skew field). Show that $Z(D)$ is a field.
\end{enumerate}

\item[\#3 Not the Same] Let $R$ and $S$ be rings with $1$.
\begin{enumerate}[(a)]
\item Show that if $R \cong S$ then $R^\times \cong S^\times$ (isomorphic rings have isomorphic groups of units).
\item Show that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
\end{enumerate}

\item[\#4 An Ideal Problem] Let $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$ be a chain of ideals in a ring $R$. Prove that $I = \bigcup_{i=1}^\infty I_i$ is an ideal of $R$.

\item[\#5 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 Prove that $(a)$ is an ideal.
\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}

\end{description}

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