Strongly π-regular elements and Drazin inverses

Abstract

We find an expression for the Drazin inverse of a strongly [Formula: see text]-regular element [Formula: see text] in the form [Formula: see text] when [Formula: see text] for some nonnegative integer [Formula: see text] and positive integer [Formula: see text]. This extends the result by Azumaya, which is the case when [Formula: see text] and [Formula: see text], and a result by Drazin, which is the case when [Formula: see text] is arbitrary and [Formula: see text]. We give new proofs of several results in the literature. For instance, we give an easy proof of the result that for two commuting Drazin invertible elements [Formula: see text], [Formula: see text] of [Formula: see text], then [Formula: see text] is Drazin invertible if and only if so is [Formula: see text], where [Formula: see text] is the Drazin inverse of [Formula: see text]. Our proof is akin to the case when [Formula: see text] and [Formula: see text] are invertible.