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

Abstract

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.

Left-star order structure of Rickart *-rings

“…A ring (R, +, ·) is called a *-ring or involutory ring if there is a bijection * : R → R such that (x * ) * = x, (x y) * = y * x * and (x + y) * = x * + y * for all x, y ∈ R. [6] If the *-ring has a unity 1, then 1 * = 1. Similarly, 0 * = 0.…”

“…Marovt et al generalized the one-sided star order from [3] for arbitrary Rickart *-rings in [6]. C¯ırulis generalized likewise in [7], the one-sided star order from [5].…”

On mappings preserving the sharp and star orders

“…Having in mind that a = lp(a)a = arp(a), we conclude that a = lp(a)b = brp(a). (Compare with [25,Theorem 1]. )…”

“…This characterization is still true for the more general context of Rickart C*-algebras, [25]. A C * -algebra is called a Rickart C*-algebra if the left annihilator (respectively, right annihilator) of any element a ∈ A is generated by a projection.…”

Relatively Orthocomplemented Skew Nearlattices in Rickart Rings

“…In [15,17,7], the notion of one-sided star order has almost simultaneously been generalized to abstract involution rings (to regular *-rings, regular Rickart *-rings and Rickart *-rings, respectively). The three approaches are compared in [8].…”