New results on (<i>b</i>, <i>c</i>)–inverses

Ke, Yuanyuan; Cvetković-Ilić, Dragana S.; Chen, Jianlong; Višnjić, Jelena

doi:10.1080/03081087.2017.1301362

Cited by 18 publications

(8 citation statements)

References 13 publications

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“…Drazin [24] pointed out that (b, c)-invertibility implies hybrid (b, c)invertibility, which further implies annihilator (b, c)-invertibility, but the reverse implications do not necessarily hold (Example 5.6 confirms this). Ke et al [38], Boasso and Kantún-Montiel [9] and Wang et al [67] proved that the reverse implications hold, for instance, if b and/or c are regular, or cab is regular. Wang et al [67] also provided new characterizations of (b, c)-invertibility in terms of the properties of the product cab.…”

Section: Proof (A)mentioning

confidence: 99%

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Ćirić

Ignjatović

Stanimirović

2022

Preprint

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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.We show that Mary’s inverse along an element, Drazin’s (b, c)-inverse, and Bott-Duffin (e,f)-inverse of a given element are just three different ways of representing the same notion – the outer inverse of this element belonging to the prescribed Green’s ℋ-class. Hence, outer inverses belonging to the prescribed Green’s ℛ- and ℒ-classes represent extensions of (b,c)-inverses and 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. Mathematics Subject Classification (2010). 20M99, 15A09, 15A24.

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Section: Proof (A)mentioning

confidence: 99%

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Ćirić

Ignjatović

Stanimirović

2022

Preprint

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“…Firstly, we study generalized inverses of tensors over a commutative ring. This part is motivated by the work of Drazin 27‐29 and Zhu 30,31 and Rakić 14 in rings with involution. Here we relate the concept of a well‐supported element in a ring with involution and study the group inverse of tensors along with the class of

(b, c)

‐inverses in a commutative ring.…”

Section: Introductionmentioning

confidence: 99%

Computation of generalized inverses of tensors via t‐product

Behera

Sahoo

Mohapatra

et al. 2021

Numerical Linear Algebra App

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Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In this paper, we study different generalized inverses of tensors over a commutative ring and a noncommutative ring. Several numerical examples are provided in support of the theoretical results. We also propose algorithms for computing the inner inverses, the Moore–Penrose inverse, and weighted Moore–Penrose inverse of tensors over a noncommutative ring. The prowess of some of the results is demonstrated by applying these ideas to solve an image deblurring problem.

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“…The following notations a R = {ax | x ∈ R}, Ra = {xa | x ∈ R} and [a, b] = ab − ba will be used in the sequel for a, b ∈ R. In [ If such y exists, then it is unique, denoted by a (b,c) , and said to be the (b, c)-inverse of a. Many existence criteria and properties of the (b, c)-inverse can be found in, for example, [3,4,9,10,13,14,19,[21][22][23]. In [10,Definition 1.2] and [14,Definition 2.1], the authors independently introduced the one-sided (b, c)-inverses in rings.…”

Section: Introductionmentioning

confidence: 99%