Coherent and precoherent elements in Rickart *-rings

We introduce the notions of coherent and precoherent elements in a Rickart *-ring, generalizing this concept from the ring of bounded operators on a Hilbert space. Some interesting properties of such elements are demonstrated, resembling those of bounded operators, e.g. the range additivity and the parallel summation. As an application, we solve some problems regarding the star partial order on Rickart *-rings.