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Let R be a unitary ring and a,b ∈ R with ab = 0. We find the 2/3 property of Drazin invertibility: if any two of a, b and a+b are Drazin invertible, then so is the third one. Then, we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units. As applications, the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism, and the existence and expression of the inverse along a product of two regular elements are obtained, which generalizes the main result of Mary and Patrício (2016) about the group inverse of a product. [ABSTRACT FROM AUTHOR]

Let R be a unital ∗ -ring. For any a , w , b ∈ R , we apply the w -core inverse to define a new class of partial orders in R , called the w -core partial order. Suppose that a , b ∈ R are w -core invertible. We say that a is below b under the w -core partial order if a w ◯ # a = a w ◯ # b and a w a w ◯ # = b w a w ◯ # , where a w ◯ # denotes the w -core inverse of a. Characterizations of the w -core partial order are given, and its relationships with several types of partial orders are also considered. In particular, we show that the core partial order coincides with the a -core partial order, and the star partial order coincides with the a ∗ -core partial order. [ABSTRACT FROM AUTHOR]

We provide existence criteria and characterizations for outer inverses in a semigroup belonging to the prescribed Green's R -, L - and H -classes. These results generalize the well-known problem of finding outer inverses of a matrix over a field with the prescribed range or/and null space. Outer inverses belonging to the prescribed Green's R - and L -classes also represent extensions of Drazin's (b, c)-inverses and Mary's inverses along an element. We provide an overview of other such extensions that have emerged recently and compare them with the extensions introduced in this paper. [ABSTRACT FROM AUTHOR]

Let R be a unital *-ring. For any a, s, t, v, w ∈ R we define the weighted w-core inverse and the weighted dual s-core inverse, extending the w-core inverse and the dual s-core inverse, respectively. An element a ∈ R has a weighted w-core inverse with the weight v if there exists some x ∈ R such that awxvx = x, xvawa = a and (awx)* = awx. Dually, an element a ∈ R has a weighted dual s-core inverse with the weight t if there exists some y ∈ R such that ytysa = y, asaty = a and (ysa)* = ysa. Several characterizations of weighted w-core invertible and weighted dual s-core invertible elements are given when weights v and t are invertible Hermitian elements. Also, the relations among the weighted w-core inverse, the weighted dual s-core inverse, the e-core inverse, the dual f-core inverse, the weighted Moore-Penrose inverse and the (v, w)-(b, c)-inverse are considered. [ABSTRACT FROM AUTHOR]

Let a , d be two elements in rings and a ∥ d be the inverse of a along d. When a ∥ d exists, we obtain several characterizations for the invertibility of a a ∥ d − a ∥ d a , which is related to the invertibility of elements expressed by certain functions of a , d and suitable elements from the center of the ring. On the other hand, some equivalent conditions for the equality a a ∥ d = a ∥ d a , as the complement of the previous invertibility in some sense, are given by means of the group inverses and the ring units, respectively. Then, the results obtained are applied in a ∗ -ring, namely, when d = a ∗ , the co-EP and EP properties are deduced correspondingly. [ABSTRACT FROM AUTHOR]

Let a,d be two elements in rings and a∥d be the inverse of a along d. When a∥d exists, we obtain several characterizations for the invertibility of aa∥d−a∥da, which is related to the invertibility of elements expressed by certain functions of a,d and suitable elements from the center of the ring. On the other hand, some equivalent conditions for the equality aa∥d=a∥da, as the complement of the previous invertibility in some sense, are given by means of the group inverses and the ring units, respectively. Then, the results obtained are applied in a ∗-ring, namely, when d=a∗, the co-EP and EP properties are deduced correspondingly.

We extend characterizations of inverses along an element to (b, c)-inverses. It is proved that if an element a in a semigroup S is both (b, c)-invertible and (c, b)-invertible for some b , c ∈ S , then abac , acab , caba , baca are group invertible, and bac (abac) ♯ = b a (caba) ♯ c = b (acab) ♯ a c = (baca) ♯ b a c is the (b, c)-inverse of a, cab (acab) ♯ = c a (baca) ♯ b = c (abac) ♯ a b = (caba) ♯ c a b is the (c, b)-inverse of a. Moreover, we establish several criteria for the existence of (b, c)-inverses and (c, b)-inverses by means of units in a ring. As applications, some new representations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given. [ABSTRACT FROM AUTHOR]

We first prove that if a is both left (b, c)-invertible and left (c, b)-invertible, then a is both (b, c)-invertible and (c, b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2019.1679073], under the conditions (ab)* = ab and (ac)* = ac. In addition, we consider that ba is (c, b)-invertible, and at the same time ca is (b, 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. Patrício [Mediterr J. Math., 2017, 14: 208] to (b, c)-inverses. As applications, we obtain that under condition (a2)* = a2, a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible. [ABSTRACT FROM AUTHOR]

In this article, some characterizations of one-sided generalized inverses are investigated. Let R be a *-ring and It is shown that for a* is right invertible if and only if In particular, the expression of a† are given whenever Moreover, it is also proven that if and only if is right invertible along a if and only if is right invertible along a, where n is an arbitrary positive integer. Finally, we present some characterizations of by using projections and units. [ABSTRACT FROM AUTHOR]

In this paper, we give new existence criteria for the inverse along an element by means of Drazin inverses, one-sided annihilator ideals, idempotents and one-sided invertibility of certain elements in semigroups and rings, respectively. In addition, we take advantage of idempotents and one-sided principal ideals to characterize the inner inverse along an element in a ring. Furthermore, their expressions are shown. Next, we obtain Cline’s formula for the inverse along an element. Finally, we study the inverse of the product along an element in a semigroup. In particular, some results in this paper recover the consequences of the classical generalized inverses such as group inverse, Drazin inverse, and Moore-Penrose inverse. [ABSTRACT FROM AUTHOR]

Let A be an n × n complex matrix. The ( B , C ) -inverse A ∥ ( B , C ) of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to A ∥ ( B 1 , C 1 ) and B ∥ ( B 2 , C 2 ) of A and B are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained.

Let be a semigroup and . The concept of -inverses was introduced by Drazin in 2012. It is well known that the Moore-Penrose inverse, the Drazin inverse, the Bott-Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the -inverse. In this paper, a new relationship between the -inverse and the Bott-Duffin -inverse is established. The relations between the -inverse of and certain classes of generalized inverses of and , and the -inverse of are characterized for some , where . Necessary and sufficient conditions for the existence of the -inverse of a lower triangular matrix over an associative ring are also given, and its expression is derived, where are regular triangular matrices. [ABSTRACT FROM AUTHOR]

The aim of this short note is to indicate that Rao and Mitra's constrained matrix inverse A c r C R (introduced in 1972) is a direct precursor of recently introduced Drazin's ( b , c ) outer generalized inverse. Precisely, if one rewrites definition of A c r C R in purely multiplicative language, then it turns out that two inverses coincide. Recently introduced Mary's inverse along an element and Manjunatha Prasad and Mohana's core-EP inverse are the special cases of Rao and Mitra's constrained inverse. [ABSTRACT FROM AUTHOR]

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a new notion in a semigroupas an extension of Mary’s inverse. Let. An elementis called left (resp. right) invertible alongif there existssuch that(resp.) and(resp.). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right)-regularity and left (right)-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary’s inverse along a matrix is expressed. [ABSTRACT FROM PUBLISHER]

Let R be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ⁎-ring case. It is shown that the group, Moore–Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in R are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of ( b , c ) -inverses. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions. [Copyright &y& Elsevier]

Let A be an n × n complex matrix. The (B , C) -inverse A ∥ (B , C) of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to A ∥ (B 1 , C 1) and B ∥ (B 2 , C 2) of A and B are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained. [ABSTRACT FROM AUTHOR]