Idempotent elements in a ring

#ring_theory

Let $R$ be a commutative ring with 1 and suppose $e \in R$ is idempotent, i.e., satisfies $e^{2} = e$ .
(a) Prove that $1 - e$ is also idempotent.
(b) Suppose $e \neq 0, 1$ . Show that $R e$ and $R (1 - e)$ are proper ideals of $R$ .
(c) Prove there is an isomorphism $R ≅ R e \times R (1 - e)$ .

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Let $R$ be a commutative ring with 1 and suppose $e\in R$ is {\bfseries idempotent}, i.e., satisfies $e^2=e$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Suppose $e\neq 0, 1$. Show that $Re$ and $R(1-e)$ are proper ideals of $R$.
	\item Prove there is an isomorphism $R\cong Re\times R(1-e)$.
\end{enumerate}