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 sucient condition for the Drazin invertibility of a - b is considered under the condition of ab = ?ba in algebras over an arbitrary field. Moreover, we give explicit representations of (a + b)D, as a function of a,b,aD and bD, whenever a3b = ba and b3a=ab.