If $e_1,\dots,e_n\in R$ are idempotent then $\langle e_1,\dots,e_n\rangle=\langle d\rangle$ for some idempotent $d\in R$

Question

Let $R$ be a commutatuive ring with a unit and let $e_1,...,e_n\in R$ be idempotent. Prove that $I=\langle e_1,...,e_n\rangle $ and generated by an idempotent element.

If $I=\langle d \rangle$ for some $d\in R$ I tries to look at $d=r_1e_1+...+r_ne_n$ and squre it but I didn't really figure it up.

Answer 1

Use induction. For $n=2$, consider $d=e_1+e_2-e_1e_2$ and note that $e_1d=e_1$, $e_2d=e_2$.