On partial orders in Rickart rings

Abstract

We consider the generalized concept of order relations in Rickart rings and Rickart -rings which was proposed by Šemrl and which covers the star partial order, the left-star partial order, the right-star partial order and the minus partial order. We show that on Rickart rings the definitions of orders introduced by Jones and Šemrl are equivalent. We also connect the generalized concept of order relations with the sharp order and prove that the sharp order is a partial order on the subset of elements in a ring with identity which have the group inverse. Properties of the sharp partial order in are studied and some known results are generalized.