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

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 $a^{2}b=aba$ and $b^{2}a=bab$, it is shown that $ab$ is Drazin invertible and that $a+b$ is Drazin invertible if and only if $1+a^{D}b$ 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.