Jacobson’s Lemma for Drazin Inverses

If a ring element α = 1 − ab is Drazin invertible, so is β = 1 − ba. We show that the Drazin inverse of β is expressible by a simple formula (in generalization of “Jacobson’s Lemma”) in terms of the Drazin inverse and the spectral idempotent of α. The commutation rules αa = a β and b α = β b are extended to the Drazin inverses, and the spectral idempotents of α and β are shown to be isomorphic. In a related direction, we also prove that Jacobson’s Lemma holds for π-regular elements and unit π-regular elements in arbitrary rings.