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)bb D , I + a D b, aa D (a + b) and aa D (a + b)bb D and find the expressions of (a + b) D , (a + b)bb D D , (I + a D b)D , etc.