Let $R$ be a ring and $a,b,c\in R$. We give a novel characterization of group inverses (resp. EP elements) by the properties of right (resp. left ) $c$-regular inverses of $a$ and discuss the relation among the strongly left $(b,c)$-invertibility of $a$, the right $ca$-regularity of $b$, and the $(b,c)$-invertibility of $a$. Finally, we investigate the sufficient and necessary condition for a ring to be a strongly left min-Abel ring by means of the $(b,c)$-inverse of $a$.