Research Article

Pseudo core inverses of a sum of morphisms

Published in: Quaestiones Mathematicae
Volume 44 , issue 10 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 1321–1332
DOI: 10.2989/16073606.2020.1791275
Author(s): Jianlong Chen , China, Wende Li , China, Mengmeng Zhou , China,
Keywords: 15A09, 16W10, 18B99, 15A09, 16W10, 18B99,
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Abstract

Let be an additive category with an involution ∗. Suppose that both ϕ : X → X is a morphism of with pseudo core inverse ϕ and η : X → X is a morphism of such that 1 + ϕ η is invertible. Let α = (1 + ϕ η) − 1 , β = (1+ηϕ ) − 1 , ε = (1−ϕϕ )ηα(1−ϕ ϕ), γ = α(1−ϕ ϕ)ηϕ β, σ = αϕ ϕα− 1(1− ϕϕ )β, δ = β∗ (ϕ ) ∗η∗ (1−ϕϕ )β. Then we present a sufficient condition such that f = ϕ + η − ε has pseudo core inverse and give the corresponding expression. Let R be a unital ∗-ring and J(R) its Jacobson radical. If a is pseudo core invertible with the pseudo core inverse a and j ∈ J(R), we also give a sufficient condition which ensures that a + j − ε has pseudo core inverse. Thus, these results generalize recent results on core inverse.
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