Revisiting generalized inverses of a sum with a radical element

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This is a Quantitative Study study.

Yukun Zhou; Néstor Thome

doi:10.48550/arxiv.2510.00901

October 1, 2025Open Access

Revisiting generalized inverses of a sum with a radical element

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Key Points

Conditions for $(b+j_b,c+j_c)$-invertibility are established, aiding in understanding matrix behaviors.
Necessary and sufficient conditions are outlined for the invertibility of $a+j_a$ with generalized inverses.
The study connects generalized inverse theory to the properties of dual matrices, enriching existing mathematical frameworks.
Highlights the role of Jacobson radical in the generalized invertibility context, expanding theoretical knowledge.

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Abstract

Let R be a ring with identity and $J(R)$ be its Jacobson radical. Assume that a∈ R is $(b,c)$-invertible and jₐ,jb,jc∈ J(R). This paper provides necessary and sufficient conditions for a+jₐ to be (b+jb,c+jc)-invertible. As an application, corresponding results on generalized inverses of dual matrices are derived.

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Zhou et al. (2025) studied this question.

synapsesocial.com/papers/68e25378d6d66a53c247429d https://doi.org/10.48550/arxiv.2510.00901

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Revisiting generalized inverses of a sum with a radical element

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Revisiting generalized inverses of a sum with a radical element

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