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Let $R = L_1 \oplus L_2 \oplus ··· \oplus L_n$ where each $L_i$ is a left ideal and $n \in \mathbb{N}$. Show that there exist idempotents $e_i \in R$ where $e_i = e_i^2$ such that $1_R = \sum\limits_{i=1}^{n} e_i$, $L_i = Re_i$ and $e_i e_j = \delta_{i,j}e_i$ for all i, j.

All I really know that is the elements $\{e_i\}$ are called orthogonal idempotents. I am looking for hints on how to proceed not a premade solution.

How would I also start to prove a variant of this where $e_i \in Z(R)$?

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  • $\begingroup$ If you have a decomposition into left ideals there is not necessarily a "variant where $e_i\in Z(R)$". Take, for instance, $M_n(F)$ for $F$ a field. You can decompose $1$ into $n$ pairwise orthogonal idempotents, but the ring only has two central idempotents. But if you have decomposed the ring into a finite "direct sum of ideals" then yes, you can find central idempotents. $\endgroup$
    – rschwieb
    Commented Aug 5, 2018 at 2:22

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Start by decomposing $1_R$ as a sum $\sum_{i=1}^ne_i$, where $e_i\in L_i$. This decomposition is unique, so you can actually be sure that these $e_i$'s must be the ones you want. Then, proceed to showing that they satisfy the conditions. You may deduce relations between the $e_i$'s by using the fact that $1_R$ is idempotent. Can you conclude with this?

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