Orthogonal idempotent ($e_i^2=e_i,e_i{e}_j=e_j{e}_i=0(i\ne j)$)

Distinct idempotents are orthogonal when their products in both orders vanish. A finite family summing to one gives a decomposition $M={⨁}_i{e}_{i}M$ of every left module as a vector space; the summands need not be submodules unless the idempotents are central. Vertex idempotents of a path algebra supply the spaces of its quiver representation.