Suitable elements, *-clean elements and Sylvester equations in rings with involution

Let R be a *-ring. In this article, we introduce good *-clean elements and left *-suitable elements in R, respectively. An element a is an element of R is good *-clean if it can be written as the sum of a projection and a unitary element. An element a is an element of R is left *-suitable if there exists a projection p is an element of Ra such that 1-p is an element of R(1-a). Several properties of them are given and their characterizations are derived by the solvability of the Sylvester equation XA-BX=C in a ring. Finally, we use generalized inverses to give the existence criterion of left *-suitable elements.