Drazin–Moore–Penrose invertibility in rings

Patrício, Pedro; Puystjens, Roland

doi:10.1016/j.laa.2004.04.006

Cited by 50 publications

(39 citation statements)

References 19 publications

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“…Let a ∈ R D with i(a) = m. The sum a = c a +n a is called the core nilpotent decomposition [14] of a, where c a = aa D a is the core part of a, and n a = (1 − aa D )a is the nilpotent part of a. This decomposition brings n m a = 0, c a n a = n a c a = 0, and c a ∈ R # with c # a = a D [14].…”

Section: General Results On Pseudo Core Inversesmentioning

confidence: 99%

Pseudo core inverses in rings with involution

Gao

Chen

2017

Communications in Algebra

126

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Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary * -ring case by Rakić, Dinčić and Djordjević in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of * -ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.

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Section: General Results On Pseudo Core Inversesmentioning

confidence: 99%

Pseudo core inverses in rings with involution

Gao

Chen

2017

Communications in Algebra

126

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“…Patricio and Puystjens [15] introduced the notions of *-EP and *-gMP in rings with involution. They said that a is *-EP if aR = a * R ; a is *-gMP if a † and a # exist with a † = a # .…”

Section: (A) It Is Clear That Ifmentioning

confidence: 99%

Characterizations of *-DMP matrices over rings

Gao¹,

Chen²

2018

Turk J Math

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Let R be a ring with involution * . R m×n denotes the set of all m × n matrices over R . In this paper, we give a characterization of the pseudo core inverse of A ∈ R n×n in the form ofThen we obtain necessary and sufficient conditions for A ∈ R n×n , in the form of A = GDH , Nr(G) = 0 , N l (H) = 0 , D 2 = D = D * , to be *-DMP. If R is a principal ideal domain (resp. semisimple Artinian ring), then matrices expressed as that form include all n × n matrices over R .

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“…It is known [15] (see also [16]) that the Moore-Penrose inverse a † of a von Neumann regular element a in an involutory ring can be characterized by the invertibility of the element a * a + 1 − ca. Moreover, Let z ∈ (a * ) • .…”

Section: Definition 12 Letmentioning

confidence: 99%

Star, left-star, and right-star partial orders in Rickart ∗-rings

Marovt

Rakić

Djordjević

2014

Linear and Multilinear Algebra

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Let A be a unital ring admitting involution. We introduce an order on A and show that in the case when A is a Rickart * -ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart * -rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart * -rings and some known results are generalized. We found matrix forms of elements a and b when a ≤ b, where ≤ is one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated.

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Drazin–Moore–Penrose invertibility in rings

Cited by 50 publications

References 19 publications

Pseudo core inverses in rings with involution

Pseudo core inverses in rings with involution

Characterizations of *-DMP matrices over rings

Star, left-star, and right-star partial orders in Rickart ∗-rings

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