  Given a unital ring $R$ and a two-sided ideal $I$ of $R$, we consider the
question of determining when a unit of $R/I$ 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*-algebras
with real rank zero.
