Idempotent elements in a ring (2)

#ring_theory

An element $r$ of a ring $R$ is said to be idempotent if $r^{2} = r$ . Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$ .

Prove that $1 - e$ is also idempotent.
Prove that $e R$ and $(1 - e) R$ are both ideals in $R$ and that
$R ≅ e R \times (1 - e) R .$
Prove that if $R$ has a unique maximal ideal, then the only idempotent elements in $R$ are 0 and 1.

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L A T E X

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An element $r$ of a ring $R$ is said to be {\bfseries idempotent} if $r^2=r$. Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Prove that $eR$ and $(1-e)R$ are both ideals in $R$ and that
	\[
		R\cong eR\times (1-e)R.
	\]
	\item Prove that if $R$ has a unique maximal ideal, then the only idempotent elements in $R$ are 0 and 1.
\end{enumerate}