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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $n \\ge 1$ be an integer.\nA *complete set of orthogonal idempotents* of $R$ is a family $e_1, \\dots, e_n$ of elements of $R$ such that:\n\n1. each $e_i$ is an [[idempotent|0X4JKZXJ86]], that is, $e_i^2 = e_i$;\n2. $e_i e_j = 0$ for all $i \\ne j$;\n3. $e_1 + \\cdots + e_n = 1$.\n\n## Remarks\n\nMany authors also require each $e_i$ to be nonzero.\nExample: in $M_n(K)$ over a field $K$, the diagonal matrix units $E_{11}, \\dots, E_{nn}$ form a complete set of orthogonal idempotents.\nThe check: $E_{ii} E_{jj}$ equals $E_{ii}$ if $i = j$ and $0$ otherwise, and $E_{11} + \\cdots + E_{nn} = I_n$.\nNon-example: in $M_2(K)$, the family consisting of $E_{11}$ alone satisfies clauses 1 and 2 but not clause 3, since $E_{11} \\ne I_2$.\nFor any idempotent $e$, the pair $e, 1 - e$ is a complete set of orthogonal idempotents.\nA complete set of orthogonal idempotents gives a decomposition $R = Re_1 \\oplus \\cdots \\oplus Re_n$ of $R$ as a left module over itself.\n",
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