Nilpotent elements in a ring

#ring_theory

Let $R$ be a commutative ring with 1. We say an element $n \in R$ is nilpotent if there exists a number $k \in N$ such that $n^{k} = 0$ .

Show that if $n$ is nilpotent, then $1 - n$ is a unit.
Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.

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Let $R$ be a commutative ring with 1. We say an element $n\in R$ is {\bfseries nilpotent} if there exists a number $k\in {\bf N}$ such that $n^k=0$.
\begin{enumerate}[label=\alph*)]
	\item Show that if $n$ is nilpotent, then $1-n$ is a unit.
	\item Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.
\end{enumerate}