In this article, we investigate additive properties of the Drazin inverse of
elements in rings and algebras over an arbitrary field. Under the weakly
commutative condition of ab=λba, we show that a−b is Drazin
invertible if and only if aaD(a−b)bbD is Drazin invertible. Next, we
give explicit representations of (a+b)D, as a function of a,b,aD
and bD, under the conditions a3b=ba and b3a=ab.Comment: 17 page