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\section*{Math 403/503 Spring 2024}
\textbf{Problems 4 and 5 were updated on March 4, 5:20pm.}

In this class, by
\begin{itemize}
\item a \emph{ring} $R$ we mean a \emph{ring with identity $1_R$};
\item a \emph{subring} $S\subseteq R$ we mean \emph{subring containing the identity $1_R$};
\item a \emph{ring homomorphism} $\varphi:R\to T$ we mean \emph{ring homomorphism sending $1_R$ to $1_T$}.
\end{itemize}
\subsection*{Homework 5, due March 6}
\begin{enumerate}
\item Determine which of the following sets are \textbf{rings} with respect to the usual operations of addition and multiplication. If the set is a ring, determine whether it is an \textbf{integral domain} or a \textbf{field} (or both). (\emph{Hint:} Use the subring criterion where appropriate.)
\begin{enumerate}[{\rm (a)}]
\item $7\mathbb{Z}$ 
\item $\mathbb{Z}_{18}$
\item $\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}\mid a,b\in\mathbb{Q}\}$
%\item $\mathbb{Z}[\sqrt{3}]=\{a+b\sqrt{3}\mid a,b\in \mathbb{Z}\}$
\item $R=\{a+b\sqrt[3]{3}\mid a,b\in\mathbb{Q}\}$
\item $\mathbb{Z}[i]=\{a+bi\mid a,b\in \mathbb{Z}\}$ where $i^2=-1$
\item $\mathbb{Q}(\sqrt[3]{3}) = \{a+b\sqrt[3]{3}+c\sqrt[3]{9} \mid a,b,c \in \mathbb{Q}\}$
\end{enumerate}
\item Find all the units in the ring $M_2(\mathbb{Z}_2)$ of all $2\times 2$-matrices with entries from $\Z_2$.
\item If $R$ and $S$ are rings, prove that $R\times S$ is a ring with component-wise operations.
\item Find all ring homomorphisms $\varphi:\mathbb{Z}/6\mathbb{Z} \to \mathbb{Z}/15\mathbb{Z}$ \textbf{or prove that there are none}.
\item Define a map $\varphi:\C\to M_2(\R)$ by
\[\varphi(a+bi)=\begin{bmatrix}a&-b\\b&a\end{bmatrix}\]
Prove that $\varphi$ is an \textbf{injective homomorphism} of rings.
\item Let $\varphi:R\to S$ be a ring homomorphism. Prove each of the following statements.
\begin{enumerate}[{\rm (a)}]
\item If $R$ is a commutative ring, then $\varphi(R)$ is a commutative subring of $S$.
\item If $R$ is a field, and $\varphi(R)\neq \{0_S\}$, then $\varphi$ is injective.
\end{enumerate}
\item Let $R$ be a ring. Define the \emph{center} of $R$ to be
\[Z(R) = \{a \in R \mid ar = ra \text{ for all $r \in R$}\}.\]
Prove that $Z(R)$ is a commutative subring of $R$.
\end{enumerate}
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