Lifting idempotents

Summary

Lifting idempotents is the question of whether an idempotent of a quotient ring $R/I$ comes from an idempotent of $R$. When every idempotent of $R/I$ lifts, decompositions of $R/I$ into pieces can be pulled back to decompositions of $R$. Idempotents always lift modulo a nil ideal, and they lift modulo the Jacobson radical exactly in semiperfect rings among the semilocal rings. The technique is central in the theory of finite-dimensional algebras, complete local rings and modular representation theory.