Complete set of orthogonal idempotents

Remarks

Many authors also require each $e_i$ to be nonzero. Example: in $M_n(K)$ over a field $K$, the diagonal matrix units $E_{11}, \dots, E_{nn}$ form a complete set of orthogonal idempotents. The check: $E_{ii} E_{jj}$ equals $E_{ii}$ if $i = j$ and $0$ otherwise, and $E_{11} + \cdots + E_{nn} = I_n$. Non-example: in $M_2(K)$, the family consisting of $E_{11}$ alone satisfies clauses 1 and 2 but not clause 3, since $E_{11} \ne I_2$. For any idempotent $e$, the pair $e, 1 - e$ is a complete set of orthogonal idempotents. A complete set of orthogonal idempotents gives a decomposition $R = Re_1 \oplus \cdots \oplus Re_n$ of $R$ as a left module over itself.