Star, sharp, core and dual core partial order in rings with involution

Partial orders in rings based on generalized inverses – unified theory

“…For the proof in the matrix case see [22] and [1]. For the proof in the ring case see [10] and [29]. In contrast to existing proofs we will prove these facts using Corollary 3.8.…”

“…The minus and star partial orders are originally defined on a semigroup and others on the set of complex matrices. In [27] and [29] these orders are considered in the context of arbitrary ring with involution. It is shown that these orders are partial orders on appropriate subsets of R. One can notice an obvious similarity in definitions of partial orders based on generalized inverses.…”

Core partial order in rings with involution

“…Baksalary and Trenkler gave several characterizations and various relationships between the matrix core partial order and other matrix partial orders by using the decomposition of Hartwig and Spindelböck [4]. In [11], Rakić and Djordjević generalized the matrix core partial order to the ring case. They gave various equivalent conditions of core partial order and investigated relationships between the core partial order and other partial orders in general rings.…”

“…Let us start with a auxiliary lemma. In [11,Theorem 4.10], Rakić and Djordjević gave the relationship between the core partial order and the minus partial order for a, b ∈ R # . For instance, it is proved that a # ≤ b if and only if a − ≤ b and b # ab # = a # .…”

“…By Lemma 3.1, the core partial order implies the left star partial order and the right sharp partial order. Motivated by [11,Theorem 4.10], we have the following theorem.…”