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  "body": "\n## Summary\n\nA complete set of orthogonal idempotents in a [[ring|0B1261B5ED]] is a finite family of [[idempotent elements|0N5NJZKKN3]].\nTheir pairwise products are zero and their sum is the unit element.\nSuch a family cuts the ring into pieces.\nAs a left module over itself, the ring is the direct sum of the left ideals generated by the idempotents.\nIt is the algebraic form of splitting a space into complementary parts by projections.\nWhen the idempotents are central, the ring itself is a direct product of rings.\n\n## Explanations\n\n### L1 human\n\nThink of projections onto the coordinate axes of $\\R^n$.\nEach one, applied twice, gives the same result as applied once.\nApplying one and then another gives zero, and together they add up to the identity.\nIn the ring of $2 \\times 2$ matrices, the diagonal matrices $E_{11} = \\mathrm{diag}(1, 0)$ and $E_{22} = \\mathrm{diag}(0, 1)$ form such a family.\n",
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