Existence criteria and expressions of the (b,c)-inverse in rings and its applications

Existence criteria for the $(b,c)$-inverse are given.% in terms of annihilators. We present explicit expressions for the $(b,c)$-inverse by using inner inverses. We answer the question when the $(b,c)$-inverse of $a\in R$ is an inner inverse of $a$. As applications, we give a unified theory of some well-known results of the $\{1,3\}$-inverse, the $\{1,4\}$-inverse, the Moore-Penrose inverse, the group inverse and the core inverse.