1,150 research outputs found
Additive Property of Drazin Invertibility of Elements
In this article, we investigate additive properties of the Drazin inverse of
elements in rings and algebras over an arbitrary field. Under the weakly
commutative condition of , we show that is Drazin
invertible if and only if is Drazin invertible. Next, we
give explicit representations of , as a function of
and , under the conditions and .Comment: 17 page
A new class of partial orders
Let be a unital -ring. For any , we apply the defined
-core inverse to define a new class of partial orders in , called the
-core partial order. Suppose are -core invertible. We say that
is below under the -core partial order, denoted by
, if and , where
denotes the -core inverse of . Characterizations of the -core partial
order are given. Also, the relationships with several types of partial orders
are considered. In particular, we show that the core partial order coincides
with the -core partial order, and the star partial order coincides with the
-core partial order
A note on clean elements and inverses along an element
Let R be an associative ring with unity 1 and let a, d is an element of R. An element a is an element of R is called invertible along d if there exists unique a(parallel to d) such that a(parallel to d) ad = d = daa(parallel to d) and a(parallel to d )is an element of dR boolean AND Rd (see [6, Definition 4]). In this note, we present new characterizations for the existence of a(parallel to d )by clean decompositions of ad and da. As applications, existence criteria for the Drazin inverse and the group inverse are given.The authors are highly grateful to the referee for his/her valuable comments and suggestions which greatly improved this paper. This research is supported by the National Natural Science Foundation of China (No. 11801124), the Fundamental Research Funds for the Central Universities (No. JZ2018HGTB0233) and the Natural Science Foundation of Anhui Province (No. 1808085QA16), and the Portuguese Funds through FCT-'Fundacao para a Ciencia e a Tecnologia', within the project UID-MAT-00013/2013
The Moore-Penrose inverse of differences and products of projectors in a ring with involution
In this paper, we study the Moore–Penrose inverses of differences and products of projectors in a ring with
involution. Some necessary and sufficient conditions for the existence of the Moore–Penrose inverse are given. Moreover, the expressions of the Moore–Penrose inverses of differences and products of projectors are presented.Portuguese Funds through FCT - ‘Fundação para a Ciência e Tecnologia’, within the project PEst-OE/MAT/UI0013/2014.info:eu-repo/semantics/publishedVersio
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