Moore-Penrose inverse is one of development of generalized inverse. In this paper, we defined and studied a relation between the Moore-Penrose inverse and the Drazin inverse in the setting of rings with involution. The results of this paper are new characterizations of Moore-Penrose inverse by applying Drazin inverse with an algebraic proof.

In this paper, we present several new characteristics of the Moore-Penrose inverse in rings with involution. We use the concept of the Drazin inverse to build these characteristics in purely algebraic terms. We only discuss on the symmetric element of Moore-Penrose invertible.