Abstract
0truemm0truemm We study properties of elements in a ring which admit the generalized Drazin inverse. It is shown that the element 1- ab is generalized Drazin invertible if and only if so is 1- ba and a formula for the generalized Drazin inverse of 1- ba in terms of the generalized Drazin inverse and the spectral idempotent of 1- ab is provided. Further, recent results relating to the Drazin index can be recovered from our theorems.
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