Idempotent refinement theorem

In a finite-dimensional algebra, or a complete finite coefficient-ring algebra, primitive idempotents lift modulo a nilpotent ideal or the coefficient maximal ideal. Orthogonal decompositions of the identity lift, and equivalent primitive idempotents have isomorphic associated projective modules. Two lifts of the same idempotent modulo such an ideal are conjugate by a unit congruent to one. This is an idempotent lifting and refinement result; it does not assert that every central idempotent lifts centrally through an arbitrary nilpotent quotient.