The equation characterizations of generalized inverses in a ring with involution

In this paper, we study the relation between the generalized inverse properties of an element in a ring with involution and related equations. Mainly, by exploring the existence of the solution in a given set and the expressions of the general solution to these constructed related equation, we obtain a lot of new characterizations of EP elements, partial isometries, SEP elements, Hermitian elements and normal elements.