标题： 重新审视带有根元素的和的广义逆 显示英文标题

标题： Revisiting generalized inverses of a sum with a radical element

摘要： 设$R$是一个含单位元的环，$J(R)$是其Jacobson根。 假设$a\in R$是$(b,c)$-可逆的且$j_a,j_b,j_c\in J(R)$。 本文提供了$a+j_a$成为$(b+j_b,c+j_c)$-可逆的必要充分条件。 作为应用，得到了对偶矩阵广义逆的相应结果。 显示英文摘要

摘要： 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.