Janko Marovt

Star, left-star, and right-star partial orders in Rickart ∗-rings

Let be a unital ring admitting involution. We introduce an order on and show that in the case when 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 and when , 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.

On partial orders in Rickart rings

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.

Minus partial order in Rickart rings

The minus partial order is already known for complex matrices and bounded linear operators on Hilbert spaces. The notion is extended to Rickart rings and it is proved that this relation is a partial order. Some well-known results are generalized.

A note on partial orders of Hartwig, Mitsch, and Šemrl

We show that on Rickart rings the partial orders of Mitsch and Semrl are equivalent. In particular, these orders are equivalent on B(H), the algebra of all bounded linear operators on a Hilbert space H.

Orders in rings based on the core-nilpotent decomposition

ABSTRACT We study orders in unital rings that are derived from the core-nilpotent decomposition. The notion of the Drazin order, the C-N partial order and the S-minus partial order is extended from , the set of all matrices over a field , to the set of all Drazin invertible elements in rings with identity. Properties of these orders are investigated, their characterizations are presented, and some known results are thus generalized.

On sets of elements in Rickart rings induced by partial orders

Abstract 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 Mosic et al. on partial orders in B ( H ) , the algebra of all bounded linear operators on a Hilbert space H , are thus generalized.

Homomorphisms of Matrix Semigroups over Division Rings from Dimension Two to Three

Let 𝔻 be an arbitrary division ring and Mn(𝔻) the multiplicative semigroup of all n × n matrices over 𝔻. We describe the general form of non-degenerate homomorphisms from M2(𝔻) to M3(𝔻).