Idempotent lifting

If a ring is complete with respect to an ideal contained in its Jacobson radical, every idempotent in the quotient lifts to an idempotent of the ring. Consequently direct-sum decompositions of a modular reduction lift to a complete integral lattice.

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.