Jacobson's Lemma for Drazin Inverses

摘要

If a ring element a = 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 a. The commutation rules aa = aβ and ba = βb are extended to the Drazin inverses, and the spectral idempotents of a 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.