On star, sharp, core, and minus partial orders in Rickart rings

Abstract

Let A be a Rickart *-ring and let ≤*,≤♯,≤⊕, and ≤⊕ denote the star, the sharp, the core, and the dual core partial orders in A, respectively. The sets of all b∈A such that a≤b, along with the sets of all b∈A such that b≤a, are characterized, where a∈A is given and where ≤ is one of the partial orders: ≤*, or ≤♯, or ≤⊕, or ≤⊕. Such sets of elements that are above or below a given element under the minus partial order ≤− in a Rickart ring A are also studied. Some recent results of Cvetković-Ilić et al. on partial orders in B(H), the algebra of all bounded linear operators on a Hilbert space H, are thus generalized.