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Let A be a unital ring admitting involution. We introduce an order on A and show that in the case when A is a Rickart * -ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart * -rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart * -rings and some known results are generalized. We found matrix forms of elements a and b when a ≤ b, where ≤ is one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated.

The minus partial order is already known for complex matrices and bounded linear operators on Hilbert spaces. We extend this notion to Rickart rings, and thus we generalize some well-known results.

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.

Abstract. Let H be an infinite dimensional complex Hilbert space, and let B(H) be the set of all bounded linear operators on H . In the paper equivalent definitions for the left-star and the right-star partial orders on B(H) are given and bijective additive maps on B(H) which preserve the left-star or the right-star partial order in both directions are characterized.Mathematics subject classification (2010): 06A06, 15A03, 15A04, 15A86.