Given a unital ring R and a two-sided ideal I of R, we consider the question of determining when a unit of RaI can be lifted to a unit of R. For the wide class of separative exchange ideals I, we show that the only obstruction to lifting invertibles relies on a K-theoretic condition on I. This allows to extend previously known index theories to this context. Using this we can draw consequences for von Neumann regular rings and C Ãalgebras with real rank zero.