Revisiting generalized inverses of a sum with a radical element
Yukun Zhou, Nestor Thome
TL;DR
This work analyzes how perturbing a ring element by a radical $j_a\in J(R)$ interacts with the $(b,c)$-inverse. It proves necessary and sufficient conditions for the existence of the perturbed inverse $(a+j_a)^{\|(b+j_b,c+j_c)}$ whenever $a^{\|(b,c)}$ exists, and provides an explicit formula $(a+j_a)^{\|(b+j_b,c+j_c)}=(1+j_bb^+)a^{\|(b,c)}(1+ja^{\|(b,c)})^{-1}(1+c^+j_c)$, along with alternative representations and corollaries for standard inverses (Moore–Penrose, group, core, Drazin). The results extend to dual matrices, yielding corresponding expressions for $(\widehat{A})^{\|(\widehat{B},\widehat{C})}$ and linking dual perturbations to full-rank decompositions. The paper also deduces structural consequences, such as strong cleanliness in the dual setting. Overall, it unifies radical-perturbation behavior across several generalized inverses and provides practical tools for dual-matrix analysis.
Abstract
Let $R$ be a ring with identity and $J(R)$ be its Jacobson radical. Assume that $a\in R$ is $(b,c)$-invertible and $j_a,j_b,j_c\in J(R)$. This paper provides necessary and sufficient conditions for $a+j_a$ to be $(b+j_b,c+j_c)$-invertible. As an application, corresponding results on generalized inverses of dual matrices are derived.