Additive and product properties of Drazin inverses of elements in a ring

TL;DR

This paper investigates the properties of Drazin inverses for sums and products of elements in a ring, providing new formulas and conditions for their invertibility, generalizing previous results in algebra.

Contribution

It introduces new conditions and formulas for the Drazin invertibility of sums and products of elements, extending existing algebraic results.

Findings

$ab$ is Drazin invertible under certain conditions

$a+b$ is Drazin invertible iff $1+a^Db$ is Drazin invertible

Explicit formulas for $(ab)^D$ and $(a+b)^D$ are provided

Abstract

We study the Drazin inverses of the sum and product of two elements in a ring. For Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is Drazin invertible and that $a+b$ is Drazin invertible if and only if $1+a^Db$ is Drazin invertible. Moreover, the formulae of $(ab)^D$ and $(a+b)^D$ are presented. Thus, a generalization of the main result of Zhuang, Chen et al. (Linear Multilinear Algebra 60 (2012) 903-910) is given.