Idempotent elements in a ring
Let
- Prove that
is also idempotent. - Suppose
. Show that and are proper ideals of . - Prove there is an isomorphism
.
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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}