Weighted Moore-Penrose Inverses and Weighted Core Inverses in Rings with Involution

In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a,b ∈ R. Then a ∈ R with a= b if and only if （ab）^＊ = ab, ba^2 = a, and ab^2 = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

Revisitation of the Core Inverse

In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to AT,S^（2）.

Characterizations for the Core Invertibility and EP-ness Involving Projections

In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of 1−pq,1−pqp,p−pq and p−q,where p,q are projections in different settings,such as∗-rings,∗-reducing rings and C∗-algebras.Moreover,several representations for the core inverses of product,difference and sum of two generalized projections are derived.In particular,a number of examples are given to illustrate our results.

Weighted binary relations involving core-EP inverse

We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations.This relation becomes a pre-order.Then,one-sided preorders associated to the weighted core-EP inverse are given from two perspectives.Finally,we make a comparison for these two sets of one-sided weighted pre-orders.

New Results on EP Elements in Rings with Involution

In this paper,we mainly give characterizations of EP elements in terms of equations.In addition,a related notion named a central EP element is defined and investigated.Finally,we focus on characterizations of a generalized EP element,i.e.,+-DMP element.

The(b,c)-inverse in semigroups and rings with involution

We first prove that if a is both left(6,c)-invertible and left(c,b)-invertible,then a is both(6,c)-invertible and(c,6)-invertible in a*-monoid,which generalizes the recent result about the inverse along an element by L.Wang and D.Mosic[Linear Multilinear Algebra,Doi.org/10.1080/03081087.2019.1679073],under the conditions(a6)*=ab and(ac)*=ac.In addition,we consider that ba is(c,fe)-invertible,and at the same time ca is(6,c)-invertible under the same conditions,which extend the related results about Moore-Penrose inverses studied by J.Chen,H.Zou,H.Zhu,and P.Patricio[Mediterr J.Math.,2017,14:208]to(6,c)-inverses.As applications,we obtain that under condition(a2)^(*)=a^(2),a is an EP element if and only if a is one-sided core invertible,if and only if a is group invertible.