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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 1.1  \\ Fall 2019 \hfill Due September 4}}}

\large

\vspace{1cm}


\begin{exercise}
A ring $R$ such that $a^2 = a$ for all $a \in R$ is called {\em Boolean}.  Prove that every Boolean ring $R$ is commutative and that $2a = 0$ for all $a \in R$.
\end{exercise}

\begin{exercise}
Let $R$ be a ring, $S$ a set, and 
\[
R^S = \{ f : S \to R\},
\]
the set of all functions from $S$ to $R$.  Prove that $R^S$ is a ring under point-wise operations.
\end{exercise}

\begin{exercise}
If $A = \Z \oplus \Z$, prove that $\operatorname{End} A$ is not commutative.  Can you identify $\operatorname{End} A$?
\end{exercise}

\begin{exercise}
Given a ring $R$, let
\[
Z(R) = \{ a \in R \, | \, ab = ba \text{ for all } b \in R \}.
\]
Prove that $Z(R)$ is a subring of $R$, called the {\em center} of $R$.
\end{exercise}

\begin{exercise}
Let $\alpha \in \C$.  Prove that if $\alpha$ is a root of a monic quadratic polynomial with integer coefficients, then
\[
\Z \left[ \alpha \right] = \{a+b \alpha \, | \, a, b \in \Z  \}
\]
is a subring of $\C$.

\end{exercise}

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