Natural partial order on rings with involution

Abstract

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 forms a sectionally semi-complemented poset. It is proved that every interval [0,x] forms a Boolean algebra 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.