On sets of elements in Rickart rings induced by partial orders

Let A be a Rickart ring and let A(1) be the set of all regular elements in A. The set of all a∈A such that a ≤ b are characterized, where b∈A(1) is given and  ≤  is the minus partial order. In case when A is a Rickart *-ring, such sets are characterized for the diamond, the left-star, the right-star, the left-sharp, and the right-sharp partial orders. Some recent results of Mosić et al. on partial orders in B(H), the algebra of all bounded linear operators on a Hilbert space H, are thus generalized.