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 G(A) of elements in a ring A with identity which have the group inverse. Properties of the sharp partial order in G(A) are studied and some known results are generalized.

Partial orders onB(H)

“…The equivalence of the statements (iii) and (vi) in the previous lemma for the group invertible elements of the Rickart ring was proved in Theorem 12 [9].…”

“…The sharp order was considered in [9] for the elements of Rickart rings where in Theorem 14 one direction of the following lemma which give a relation between the minus and sharp partial orders was proved. Proof.…”

On partial orders in proper $*$-rings

“…Mitra introduced in [9] a partial order on the set of all n × n matrices over a field F which have the group inverse. This order, known as the sharp partial order, was generalized in [6] and independently in [13] to rings. The definition from [6] follows.…”

“…This order, known as the sharp partial order, was generalized in [6] and independently in [13] to rings. The definition from [6] follows. Denote by G(A) the set of all elements in A which have the group inverse.…”

Minus partial order and linear preservers

“…Several order relations received similar treatment (see [19], [20] and the references therein). During the last three decades several results involving linear preservers of order relations have been published.…”