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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.

The sharp order is a well known partial order defined on the set of complex matrices with index less or equal one. Following Šemrl's approach, Efimov extended this order to the set of those bounded Banach space operators A for which the closure of the image and kernel are topologically complementary subspaces. In order to extend the sharp order to arbitrary ring R (particulary to Rickart and Rickart * -rings) we use the notions of annihilators. The concept of the sharp order is extended to the set I R of those elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent. It is proved that the sharp order is a partial order relation on I R . Following the idea we also extend and discuss the recently introduced concept of core partial order.