Further results on the reverse order law for the group inverse in rings

Liu, Xiaoji; Zhang, Miao; Benítez, Julio

doi:10.1016/j.amc.2013.12.030

Cited by 10 publications

(3 citation statements)

References 13 publications

(17 reference statements)

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“…From that time, reverse order law for generalized inverses have significantly impacted many areas of science and engineering in the context of matrices [4], operators [10], tensors [24,27] and elements of rings with involution [34,23]. In particular, Koliha et al [18] discussed the reverse order law for the MoorePenrose invertible elements and Liu et al [20] derived some equivalences of the reverse order law for the group invertible elements in a ring. Further, the authors of [34] have discussed a few one-sided reverse order law and two-sided reverse order law for the core inverse in rings.…”

Section: Introductionmentioning

confidence: 99%

Further results on weighted core inverse in a ring

Das¹,

Sahoo²,

Behera³

2022

Linear and Multilinear Algebra

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The notion of the weighted core inverse in a ring with involution was introduced, recently [ Mosić et al. Comm. Algebra, 2018; 46(6); 2332-2345. In this paper, we explore new representation and characterization of the weighted core inverse of sum and difference of two weighted core invertible elements in ring with involution under different conditions. Further, we discuss reverse order laws and mixed-type reverse order laws for the weighted core invertible elements in a ring.

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Section: Introductionmentioning

confidence: 99%

Further results on weighted core inverse in a ring

Das¹,

Sahoo²,

Behera³

2022

Linear and Multilinear Algebra

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“…In general, the previous equality doesn't hold when the ordinary inverse is replaced by generalized inverse. Since the reverse order law for the generalized inverse is a useful computational tool in applications and it is significant from the theoretical point of view, many papers characterized the reverse order laws, such as [2][3][4][5][6][7][8][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Reverse order laws for Hirano inverses in rings

Chen

Zou

2019

Filomat

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In this paper, various kinds of reverse order laws for the Hirano inverse are characterized in a ring with the unit 1. Under some of conditions aba = a 2 b, ab 2 = bab, a h ab 2 = ba h ab, a 2 bb h = abb h a instead of the condition ab = ba, respectively, we present some equivalent conditions of the reverse order laws for Hirano inverses. Definition 1.1. Let a ∈ R. If there exists b ∈ R such that (2) bab = b, (5) ab = ba, (7) a 2 − ab ∈ N(R), then b is called the Hirano inverse of a, and a is Hirano invertible. Remark 1. For a ∈ R, the set of elements satisfying the (i)th equation in Definition 1.1 is denoted by a{i}, where i ∈ {2, 5, 7}. And the set of Hirano invertible elements in R is denoted by R h. Definition 1.2. Let a ∈ R. If there exists b ∈ R such that (2) bab = b, (5) ab = ba, (7) a − a 2 b ∈ N(R), then b is called the Drazin inverse of a, and a is Drazin invertible.

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“…is the generalized Schur complement of M. In addition, there are many results on the representations for group (Drazin) inverse of block matrices (operators) and their applications [10,11,17,19,22].…”

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confidence: 99%

Group invertible block matrices

2016

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Let M = A B C D (A and D are square) be a 2 × 2 block matrix over a skew field, where A is group invertible. Let S = D − CA # B denote the generalized Schur complement of M. We give the representations and the group invertibility of M under each of the following conditions: (1) S = 0; (2) S is group invertible and CA π B = 0, where A π = I − AA # . And the second result generalizes a result of C.

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Further results on the reverse order law for the group inverse in rings

Abstract: Abstract. In this paper, we use the Drazin inverse to derive some new equivalences of the reverse order law for the group inverse in unitary rings. Moreover, if the ring has an involution, we present more equivalences when both involved elements are EP.

Cited by 10 publications

References 13 publications

Further results on weighted core inverse in a ring

Further results on weighted core inverse in a ring

Reverse order laws for Hirano inverses in rings

Group invertible block matrices

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