The absorption laws for the generalized inverses

Liu, Xiaoji; Jin, Hongwei; Cvetković-Ilić, Dragana S.

doi:10.1016/j.amc.2012.08.049

Cited by 10 publications

(5 citation statements)

References 7 publications

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“…The above equality is known as the absorption law of invertible elements. In general, the absorption law does not hold for generalized inverses (see [9,10]). In this section, the absorption laws for the hybrid (b, c)-inverse are obtained.…”

Section: Absorption Law For the Hybrid (B C)-inversementioning

confidence: 99%

Further results on hybrid (b,c)-inverses in rings

Wang

2019

Filomat

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Let R be a ring and b, c ∈ R. In this paper, the absorption law for the hybrid (b, c)-inverse in a ring is considered. Also, by using the Green's preorders and relations, we obtain the reverse order law of the hybrid (b, c)-inverse. As applications, we obtain the related results for the (b, c)-inverse. Definition 1.1. Let R be an associative ring and let b, c ∈ R. An element a ∈ R is (b, c)-invertible if there exists y ∈ R such that y ∈ (bRy) ∩ (yRc), yab = b, cay = c. If such y exists, it is unique and is denoted by a (b,c). Drazin [2] also presented an equivalent characterization for the (b, c)-inverse y of a as yay = y, yR = bR and Ry = Rc. As generalizations of (b, c)-inverses, hybrid (b, c)-inverses and annihilator (b, c)-inverses were introduced in [2]. The symbols lann(a) = { ∈ R : a = 0} and rann(a) = {h ∈ R : ah = 0} denote the sets of all left annihilators and right annihilators of a, respectively. Definition 1.2. Let a, b, c, y ∈ R. We say that y is a hybrid (b, c)-inverse of a if yay = y, yR = bR, rann(y) = rann(c). If such y exists, it is unique. In this article, we use the symbol a (b c) to denote the hybrid (b, c)-inverse of a. Definition 1.3. Let a, b, c, y ∈ R. We say that y is a annihilator (b, c)-inverse of a if yay = y, lann(y) = lann(b), lann(y) = lann(c).

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Section: Absorption Law For the Hybrid (B C)-inversementioning

confidence: 99%

Further results on hybrid (b,c)-inverses in rings

Wang

2019

Filomat

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“…The equality (2.1) is known as the absorption law of invertible elements. In general, the absorption law does not hold for generalized inverses, for example, [11,15]. In this section, the absorption laws for one-sided (b, c)-inverses are obtained.…”

Section: Absorption Laws For the (B C)-inversementioning

confidence: 99%

Centralizer’s applications to the (b, c)-inverses in rings

Chen

Benítez

et al. 2018

RACSAM

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“…In general, the absorption laws for group inverses, Drazin inverses, Moore-Penrose inverses, {1,3}-inverses and {1,4}-inverses do not hold. So, many papers [1,3,6] devoted to the study of these aspects.…”

Section: Introductionmentioning

confidence: 99%

Centralizer’s applications to the inverse along an element

Zhu

Chen

Patrício

et al. 2017

Applied Mathematics and Computation

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In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements.Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution.

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The absorption laws for the generalized inverses

Cited by 10 publications

References 7 publications

Further results on hybrid (b,c)-inverses in rings

Further results on hybrid (b,c)-inverses in rings

Centralizer’s applications to the (b, c)-inverses in rings

Centralizer’s applications to the inverse along an element

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