One sided a_idempotent, one sided a_equivalent and SEP elements in a ring with involution

In order to study the properties of SEP elements, we propose the concepts of one sided a_idempotent and one sided a_equivalent. Under the condition that an element in a ring is both group invertible and MP_invertible, some equivalent conditions of such an element to be an SEP element are given based on these two concepts, as will as based on projections and the second and the third power of some products of some elements.