Order structure of a right-strong Rickart ring under the right star order

Abstract

We introduce the right star order, which is known on Rickart *-rings, in the more general setting of so-called right-strong Rickart ring s (a “star-free” generalization of Rickart *-rings). The main result is that, like a Rickart *-ring, a right-strong Rickart ring with the right star order is a relatively orthocomplemented poset. Since a right-strong Rickart ring is not necessarily involutive, this shows that the right star order can be defined and described in terms that are independent of an involution and that some of the results obtained for Rickart *-rings can be preserved.