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

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)bbD, I + aDb, aaD(a + b) and aaD(a + b)bbD and find the expressions of (a + b)D, [(a + b)bbD]D, (I + aDb)D, etc.