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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 2.2  \\ Fall 2019 \hfill Due September 11}}}

\large

\vspace{1cm}

\begin{exercise}
Let $R$ be a ring and let $\varphi_R : \Z \to R$ denote the map $n \mapsto n \cdot 1_R$.
\begin{itemize}

\item[\bf a.]  Prove that $\varphi_R$ is the {\em only} homomorphism from $\Z$ to $R$.  

\item[\bf b.]  Let $\varphi : R \to R'$ be a homomorphism of rings.  Prove that the following diagram commutes:
\[
\xymatrix{R \ar[rr]^{\varphi} & & R' \\ & \Z \ar[ul]^{\varphi_R} \ar[ur]_{\varphi_{R'}}& }
\]
\end{itemize}

\end{exercise}

\begin{exercise}
Let $R$ be a commutative ring with prime characteristic $p$.  Prove that for all $a, b \in R$ and all $n \in \N$ one has $(a+b)^{p^n} = a^{p^n} + b^{p^n}$.
\end{exercise}

\begin{exercise}
An element $a$ in a ring $R$ is called {\em nilpotent} if there is an $n \in \N$ so that $a^n = 0$.
\begin{itemize}

\item[\bf a.]  Give an example of a ring with nonzero nilpotents.

\item[\bf b.]  If $a, b \in R$ are nilpotent and $ab = ba$, prove that $a+b$ is nilpotent.

\end{itemize}
\end{exercise}

\begin{exercise}
Let $R$ be a ring.  Prove that if $a \in R$ is nilpotent, then $1 \pm a \in R^{\times}$.
\end{exercise}

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