Natural Partial Order on Rings with Involution

In this paper, we introduce a partial order on rings with involution, which is a generalization of the partial order on the set of projections in a Rickart *-ring. We prove that a *-ring with the natural partial order form a sectionally semi-complemented poset. It is proved that every interval [0,x] forms an orthomodular lattice in case of abelian Rickart *-rings. The concepts of generalized comparability (GC) and partial comparability (PC) are extended to involve all the elements of a *-ring. Further, it is proved that these concepts are equivalent in finite abelian Rickart *-rings.