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

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