Abstract
Let R be a prime ring with involution '*' and psi : R -> R be an endomorphism on R. In this article, we study the action of involution '*', and the effect of endomorphism psi satisfying [psi(x), psi(x*)] - [x, x*] is an element of Z(R) for all x is an element of R. In particular, we prove that any centralizing involution on prime rings with involution of characteristic different from two is of the first kind or R satisfies s(4), the standard polynomial identity in four variables. Further, we establish that if a prime ring R with involution of characteristic different from two admits a non-trivial endomorphism such that [psi(x), psi(x*)] - [x, x*] is an element of Z(R) for all x is an element of R, then the involution is of the first kind or R satisfies s(4) and [psi(x), x] = 0 for all x is an element of R.