On the $(b, c)$-inverse of a sum with a radical element in a ring

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 $(\widehat{B},\widehat{C})$-inverses of a dual matrix $\widehat{A}$ are derived.