Complete set of orthogonal idempotents

Summary

A complete set of orthogonal idempotents in a ring is a finite family of idempotent elements. 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.