The diamond partial order in rings

Abstract

In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran's theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157-169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyze successors, predecessors and maximal elements under the diamond order.

New matrix partial order based on spectrally orthogonal matrix decomposition

“…Example 3.9 It is not hard to see that I n 2,3 ≤ J n (1) but rk I n = rk J n (1) = n. Remark 3. 10 The above examples show that in contrast to all known extensions of the sharp order, see [4], the 2,3 ≤ -order is unrelated with the minus order and has nonstandard behaviour with respect to the rank function.…”

On partial orders in Rickart rings

“…In the rest of the section, we will generalize some of these results (see also [24]). The first lemma will be needed in the continuation.…”

Partial orders in rings based on generalized inverses – unified theory

“…But, in recent years, a number of papers was published considering the generalized inverses and associated partial orders in rings, see for example [14]. Furthermore, some new generalized inverses, such as core and dual core inverse (see [1]), (b, c)-inverse (see [4]) and an inverse along an element (see [19]), are introduced.…”