Drazin invertibility for sum and product of two elements in a ring

Abstract

In a ring, the expressions
for the Drazin inverses of the sum $a+b$ and the product $ab$ have been studied in some literature under the assumption that the two Drazin
invertible elements $a, b$ are commutative. In this paper, we will extend the known research results under the weaker conditions. Meanwhile, we characterize the relations of $a+b$, $(a+b)bb^D$, $\mathcal {I}+a^Db$, $aa^D(a+b)$ and $aa^D(a+b)bb^D$ and find the expressions of $(a+b)^D$,
$\left[(a+b)bb^D\right]^D$, $(\mathcal {I}+a^Db)^D$, etc.