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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Apr. $29^\mathrm{th}$, 2015}

\vspace{0.2in}

\begin{enumerate}
\item In each of the following rings, $ R $: state the characteristic of the ring, 
     give an example of a unit and its inverse (other than $ 1$), and give an example of a zero divisor.
     If no unit exists, explain why not. If no zero divisors exist, explain why not.
     \begin{enumerate}[(a)]
     \item $ R = \mathbb{Z}_4 \oplus \mathbb{Z}_6 $
     \item $ R = 3\mathbb{Z} = \{ 3k \,|\, k \in \mathbb{Z} \} $ (multiples of 3)
     \item $ R = (\mathbb{Z}_3)^{2 \times 2} $ ($2 \times 2$ matrices with entries in $ \mathbb{Z}_3 $)
     \end{enumerate}     
\item Recall $ R \oplus S $ is the direct product of the rings $ R $ and $ S $.
     \begin{enumerate}[(a)]
     \item Suppose $ R $ and $ S $ have 1's. Then show $ R \oplus S $ is also a ring with 1.
     \item Let $ R $ and $ S $  be rings with 1. Prove $ U(R \oplus S) \cong U(R) \oplus U(S) $ (i.e. the group of units of $ R \oplus S $ is
          isomorphic to the direct product of the group of units of $ R $ and the group of units of $ S $).
          {\it Hint:} You will need to write down a (group) isomorphism.
     \end{enumerate}
\item Let $ \displaystyle{ S = \left\{ \begin{bmatrix} x & 0 \\ x & 0 \end{bmatrix} \,\Big|\, x \in \mathbb{R} \right\} } $.
     \begin{enumerate}[(a)]
     \item Show $ S $ is a subring of $ \mathbb{R}^{2 \times 2} $ (the ring of $ 2 \times 2 $ real matrices).
     \item Is $ S $ commutative?
     \item Does $ S $ have a unity (i.e. a multiplicative identity)? If so, what is it?
     \item Is $ S $ an integral domain? A field?
     \end{enumerate}
\item Let $ R $ be a ring and let $ I $ and $ J $ be ideals in $ R $.  Note that $ I+J = \{ x+y \,|\, x \in I \mbox{ and } y \in j \} $, $ I \cap J $, and $ IJ = \{ x_1y_1+\cdots+x_ky_k \,|\, x_i \in I \mathrm{\;and\;} y_i \in J \} $ are ideals of $ R $.
     \begin{enumerate}[(a)]
     \item Prove that $ I+J = \{ x+y \,|\, x \in I \mbox{ and } y \in j \} $ is an ideal of $ R $.
     \item Consider the principle ideals $ (4) $ and $ (6) $ in $ \mathbb{Z} $. 
          What ideal do we get when we add them together: $ (4)+(6) $? Intersect: $ (4) \cap (6) $? Multiply $ (4)(6) $?
     \item Make a conjecture about the relationship between $m,n,d,\ell,p \in \mathbb{Z} $ if
          as ideals we have ``$ (m)+(n) = (d) $, $ (m) \cap (n) = (\ell) $, and $ (m)(n)=(p) $''.
     \end{enumerate}
\item Let $ I = (x^2) $. Create addition and multiplication tables for $ \mathbb{Z}_2[x]/I $.  Is this quotient ring a field?
 \end{enumerate}
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