Additive property of Drazin invertibility of elements in a ring

Abstract

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. The necessary and sufficient condition for the Drazin invertibility of $a-b$ is considered under the condition of $ab = \lambda ba$ in algebras over an arbitrary field. Moreover, we give explicit representations of $(a+b)^{D}$, as a function of $a, b, a^{D}$ and $b^{D}$, whenever $a^{3}b = ba$ and $b^{3}a = ab$