Generalized Inverses of a Sum in Rings

Castro‐González, N.; Araújo, C. Mendes; Patrício, Pedro

doi:10.1017/s0004972710000080

Cited by 43 publications

(26 citation statements)

References 9 publications

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“…In the case where α = 1 − ab is polar, the equation in part (C) relating the mock Drazin inverse of β to that of α boils down to the Drazin inverse formula for Jacobson pairs obtained in [21, Theorem 2.1], and the equation f = bera in part (E) recovers [21,Theorem 2.4]; see also [3,Theorem 3.6]. 4 In the case where α is quasipolar (resp.…”

Section: Mock Drazin Inverses For Strongly Clean Decompositionsmentioning

confidence: 76%

“…We say accordingly that Jacobson's Lemma holds for Drazin invertible elements. In [3] a formula for the Drazin inverse of β was found in terms of the Drazin inverse of α, and in [21] further expressions for the Drazin inverse were proven. Each of these formulas can be thought of as generalizations of the Desert Island Formula.…”

Section: Introductionmentioning

confidence: 94%

“…According to Theorem 3.7 above, we now also know that the same result holds for a number of other cases; for instance, if 1 − ab is strongly clean, or uniquely strongly clean, then so is 1 − ba. 3 In a way, it is no longer surprising that such results do not apply, for instance, to clean elements since clean decompositions for α = 1 − ab need not be Bott-Duffin decompositions (in the sense of Definition 2.1). Taking the "uniquely strongly clean" case, we conclude this section with the following nontrivial application of Theorem 3.7.…”

Section: Preserving More Propertiesmentioning

confidence: 99%

“…The proof of[3, Theorem 3.6] was harder and more indirect since it involved a lengthy reduction to the case where α is group-invertible.…”

mentioning

confidence: 99%

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Jacobson pairs and Bott-Duffin decompositions in rings

Lam¹,

Nielsen²

2019

Contemporary Mathematics

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We define the Bott-Duffin decompositions of elements in a ring, which generalize the strongly clean decompositions, and prove that the Bott-Duffin decompositions of 1 − ab are in a natural bijection with those of 1 − ba. This bijection respects a number of well known additive decompositions of elements in a ring. For instance, the result implies that 1 − ab is strongly clean (respectively, strongly nil-clean, Drazin invertible, quasipolar, or pseudopolar) if and only if so is 1 − ba. Examples and further applications are given. C(S) := {r ∈ R : rs = sr for all s ∈ S}.We write C 2 (S) for the double-centralizer C(C(S)), and when S = {s} is a single element, we write C(s) rather than C({s}). Proposition 1.1. Fix an element α ∈ R = End(M k ). Given x ∈ R, there exists an additive decomposition α = e + x, where e ∈ idem(R) ∩ C(α), if and only if there exists a direct sum decomposition diagramProof. (⇒): Assume α = e + x for some idempotent e 2 = e ∈ R which commutes with α. Since x = α − e, we see that e commutes with x as well. We

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Section: Mock Drazin Inverses For Strongly Clean Decompositionsmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 94%

Section: Preserving More Propertiesmentioning

confidence: 99%

“…The proof of[3, Theorem 3.6] was harder and more indirect since it involved a lengthy reduction to the case where α is group-invertible.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Jacobson pairs and Bott-Duffin decompositions in rings

Lam¹,

Nielsen²

2019

Contemporary Mathematics

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“…Jacobson's lemma [9,4] states that for any a, b ∈ R, if 1 − ab is invertible then so is 1 − ba. Jacobson's lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse [2,6], generalized Drazin inverse [11], and inner inverse [3].…”

Section: Introductionmentioning

confidence: 99%

Jacobson’s Lemma via Gröbner-Shirshov Bases

Zhao

2017

Algebra Colloq.

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Let R be a ring with identity 1. Jacobson's lemma states that for any a, b ∈ R, if 1 − ab is invertible then so is 1 − ba. Jacobson's lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba) −1 , assuming the latter exists.

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New results on common properties of the products AC and BA, II

Zeng

Yan

Zhang

2020

Mathematische Nachrichten

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In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If are elements in a unital associative ring satisfying , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal of ) of is converted into that of . (ii) If are bounded linear operators satisfying , then and share common complementability of kernels and ranges. (iii) If are elements in a unital semisimple Banach algebra satisfying and is a trace ideal of such that soc kh(soc, then and share common Fredholmness relative to and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

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Generalized Inverses of a Sum in Rings

Cited by 43 publications

References 9 publications

Jacobson pairs and Bott-Duffin decompositions in rings

Jacobson pairs and Bott-Duffin decompositions in rings

Jacobson’s Lemma via Gröbner-Shirshov Bases

New results on common properties of the products AC and BA, II

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