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\title{\textbf{Homework 11}}

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\section*{Section 16.7}
\textbf{Exercise 1.}\\
Which of the following sets are rings with respect to the usual
operations of addition and multiplication?  If the set is a ring, is
it also a field?
\begin{enumerate}
\item[(b)] $\Z_{18}$
\vskip1ex
\item[(b)] $\Z[\sqrt{3}] = \{a+b\sqrt{3} : a,b \in \Z\}$.
\vskip1ex
\item[(h)]  (bonus) $\Q[ \sqrt[3]{3}\, ] = \{ a + b \sqrt[3]{3} + c \sqrt[3]{9} : a, b,c \in \Q \}$
\end{enumerate}
\vskip1ex
\emph{Solution:}
\vskip1ex
\hfill $\square$

\vskip2ex

\textbf{Exercise 2.}\\
Let $R$ be a ring of $2 \times 2$ matrices of the form $$ \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix},$$
where $a,b \in \R$. Show that although $R$ is a ring that has no identity, we can find a subring $S$ of $R$ with an identity.
\vskip1ex
\emph{Proof:} 
\vskip1ex
\hfill $\square$

\vskip2ex
 
\textbf{Exercise 3.}\\
List or characterize all of the units in each of the following rings.
\begin{enumerate}
 \item[(b)] $\Z_{12}$
 \item[(d)] ${ \mathbb M}_2( \Z )$, the $2 \times 2$ matrices with entries in $\Z$
\end{enumerate}
\vskip1ex
\emph{Solution:}
\vskip1ex
\hfill $\square$

\vskip2ex

\textbf{Exercise 8.}\\ Prove or disprove: 
The ring $\Q[\sqrt{2}] = \{a+b\sqrt{2}: a,b \in \Q \}$ is isomorphic to the ring $\Q[\sqrt{3}] = \{a+b\sqrt{3}: a,b \in \Q \}$.
\vskip1ex
\emph{Solution:}
\vskip1ex \hfill $\square$

\vskip2ex

\textbf{Exercise 9.}\\
What is the characteristic of the field formed by the set of matrices 
$$F = \Bigg \{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \Bigg \}$$
with entries in $\Z_2$?
\vskip1ex
\emph{Solution:}
\vskip1ex
\hfill $\square$

\vskip2ex

\textbf{Exercise 18.}\\
Let $\phi: R \to S$ be a ring homomorphism. Prove the following statements: 
\begin{enumerate}
\item[(c)] Let $1_R$ and $1_S$ be the identities for $R$ and $S$, respectively. If $\phi$ is onto, then $\phi(1_R) = 1_S$.   
\item[(d)] If $R$ is a field and $\phi(R) \neq 0$ then $\phi(R)$ is a field.
\end{enumerate}
\vskip1ex
\emph{Proof:} 
\vskip1ex
\hfill $\square$

\vskip2ex

\textbf{Exercise 23.}\\
Let $R$ be a ring and $S$ a subset of $R$. Show that $S$ is a subring of $R$ if and only if each of the following conditions is satisfied: 
\begin{enumerate}
\item[(a)] $S \neq \varnothing$.
\item[(b)] $rs \in S$ for all $r, s \in S$.
\item[(c)] $r-s \in S$ for all $r, s \in S$.
\end{enumerate}
\vskip1ex
\emph{Proof:}
\vskip1ex
\hfill $\square$

\textbf{Exercise 31.}\\
If we do not require the (multiplicative) identity of a ring to be distinct from 0, we will not have a very interesting mathematical structure. Let $R$ be a ring such that $1=0$.
Prove that $R = \{0\}$.
\vskip1ex
\emph{Proof:}
\vskip1ex
\hfill $\square$

\vskip2ex

\textbf{Exercise 36.}\\
Let $R$ and $S$ be arbitrary rings. Show that their Cartesian product is a ring if we define addition and multiplication in $R \times S$ by
\begin{enumerate}
\item $(r,s)+(r',s') = (r+r',s+s')$
\item $(r,s)(r',s') = (rr',ss')$
\end{enumerate}
\vskip1ex
\emph{Proof:}
\vskip1ex
\hfill $\square$

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