Center-like subsets in rings with derivations or epimorphisms

We introduce centerlike subsets Z*(R,f), Z**(R,f) and Z1(R,f), where R is a ring and f is a map from R to R. For f a derivation or a nonidentity epimorphism and R a suitablychosen prime or semiprime ring, we prove that these sets coincide with the center of R.