---
tag: SS8T32TMEX
kind: concept
name: Complete set of orthogonal idempotents
msc: [16U40]
sources:
  - url: https://en.wikipedia.org/wiki/Idempotent_(ring_theory)
---

## Summary

A complete set of orthogonal idempotents in a [[ring|0B1261B5ED]] is a finite family of [[idempotent elements|0N5NJZKKN3]].
Their pairwise products are zero and their sum is the unit element.
Such a family cuts the ring into pieces.
As a left module over itself, the ring is the direct sum of the left ideals generated by the idempotents.
It is the algebraic form of splitting a space into complementary parts by projections.
When the idempotents are central, the ring itself is a direct product of rings.

## Explanations

### L1 human

Think of projections onto the coordinate axes of $\R^n$.
Each one, applied twice, gives the same result as applied once.
Applying one and then another gives zero, and together they add up to the identity.
In 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.