Partial orders on the power sets of Baer rings

Let R be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set P(R) of R and show that this relation, which we call "the minus order on P(R)", is a partial order when R is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer *-rings. We show that some ideals generated by projections of a von Neumann regular and Baer *-ring R. form a lattice with respect to the star partial order on P(R). As a particular case, we present characterizations of these orders on the power set of B(H), the algebra of all bounded linear operators on a Hilbert space H.